Dynamics and steady states of a tracer particle in a confined critical fluid

The dynamics and the steady states of a point-like tracer particle immersed in a confined critical fluid are studied. The fluid is modeled field-theoretically in terms of an order parameter (concentration or density field) obeying dissipative or conservative equilibrium dynamics and (non-)symmetry-breaking boundary conditions. The tracer, which represents, e.g., a colloidal particle, interacts with the fluid by locally modifying its chemical potential or its correlations. The coupling between tracer and fluid gives rise to a nonlinear and non-Markovian tracer dynamics, which is investigated here analytically and via numerical simulations for a one-dimensional system. From the coupled Langevin equations for the tracer-fluid system we derive an effective Fokker-Planck equation for the tracer by means of adiabatic elimination as well as perturbation theory within a weak-coupling approximation. The effective tracer dynamics is found to be governed by a fluctuation-induced (Casimir) potential, a spatially dependent mobility, and a spatially dependent (multiplicative) noise, the characteristics of which depend on the interaction and the boundary conditions. The steady-state distribution of the tracer is typically inhomogeneous. Notably, when detailed balance is broken, the driving of the temporally correlated noise can induce an effective attraction of the tracer towards a boundary.


I. INTRODUCTION
Fluids near critical points are characterized by fluctuations with large correlation length and long relaxation time, which give rise to a plethora of intriguing phenomena. A particular example is the critical Casimir force (CCF), which acts on objects immersed in a near-critical medium [1][2][3][4][5]. CCFs have been utilized to externally control the behavior of colloidal particles in critical solvents by, e.g., altering the solvent temperature or tuning the surface properties of the particles (see Ref. [6] for a review). Except in d = 2 spatial dimensions, where exact methods exist [7][8][9][10], the interaction between two colloidal particles or between a colloid and a solid wall is typically analyzed separately in the near-distance (Derjaguin) and the far-distance limit. In the former, an approximate parallel plate geometry is realized, allowing one to invoke known results for the CCF in a thin film [2,[11][12][13]. By contrast, in the far-distance limit, where the ratio between the particle radius and the distance is small, the CCF can be obtained from a "small-sphere expansion" of the Boltzmann weight [11,[14][15][16]. Recently, the non-equilibrium dynamics of colloidal particles in critical media has received increased attention [17,18], examples including studies of drag forces [18][19][20][21][22][23], aggregation [18,24], diffusion [25][26][27][28][29][30][31], shear flow [32,33], solvent coarsening [34][35][36], and interplay between criticality and activity [37,38].
Here, we study the behavior of a point-like tracer particle in a confined critical fluid within a dynamical field theory (see Fig. 1). The fluid medium is modeled in terms of a scalar order-parameter (OP) field φ(r) governed by non-conserved or conserved equilibrium dynamics within the Gaussian approximation [39,40]. In a single-component fluid, the OP represents the deviation of the local fluid density from its critical value (φ ∝ n−n c ), whereas, in a binary mixture, φ correspondingly represents the concentration deviation of a certain species [41]. We distinguish between a reactive and a passive tracer: a reactive tracer interacts with the fluid in a way that preserves detailed balance and that thus renders a steady state in accordance with equilibrium statistical mechanics. A passive tracer, by contrast, is affected by the fluid but does not act back on it and hence represents a non-equilibrium system. In the present case, the passive tracer is driven by a temporally correlated noise and thus can, in fact, be regarded as a special type of "active" particle [42][43][44][45]. Following Refs. [25][26][27], fluid and tracer are coupled such that, in the reactive case, one obtains either a locally enhanced mean OP or a locally suppressed OP variance (and reduced correlation length) at the tracer location. This resembles the typical behavior of a colloid in a critical fluid [1, 12,46] or of a magnetic point defect in the Ising model [47,48]. A reactive tracer can thus be regarded as a simplified model for a colloidal particle. We analyze theoretically and via simulations the resulting dynamics and long-time steady states for various symmetryand non-symmetry-breaking boundary conditions (BCs) imposed on the OP by the confinement. Specifically, we focus on a one-dimensional system, i.e., an interval of length L. However, we also calculate equilibrium distributions in three dimensions, which turn out to be qualitatively similar to the one-dimensional case. By comparing with the FIG. 1. Situation considered in the present study: a point-like tracer at location R(t) (black dot) is immersed in a fluctuating medium described by a scalar field φ(z, t) (order parameter). The system is confined by boundaries at z = 0, L, which exert reflective BCs on the tracer and either periodic, Neumann, Dirichlet, or capillary BCs on the field φ [see Eq. (2. 3)].
literature on CCFs, we show that the equilibrium distributions obtained here encode the static behavior of a colloid in a half-space geometry in the far-distance limit [11,14,15]. The point-like representation of the tracer thus retains the essential character of the interaction with the medium, while it facilitates analytical and numerical approaches.
The present study is structured as follows: In Secs. II and III, we introduce the model and provide the necessary preliminaries. In Secs. IV and V, the statics and dynamics of a passive and a reactive tracer, respectively, are analyzed. Results of numerical simulations are presented in Sec. VI, together with a physical interpretation of the observed phenomena and a discussion in the context of existing literature. Sec. VII provides a summary of our study. Technical details are collected in Appendices A to H.

II. MODEL
Before specializing to the one-dimensional case, we introduce the model for arbitrary spatial dimension d. We consider a system consisting of a tracer at position R(t) and a fluctuating OP field φ(r, t). The system is finite in at least one direction, which we take to be the z-direction, having length L. In thermal equilibrium, the system is described by the joint steady-state probability distribution  [49]. Tracer and OP field are coupled either linearly (with strength h) or quadratically (with strength c) [25][26][27]. The coupling h corresponds to a local bulk field (chemical potential), which leads to an excess OP around the tracer, resembling critical adsorption on a colloid [1,12]. A quadratic coupling c > 0 describes a locally reduced correlation length and results in a suppression of the OP fluctuations near the tracer. The dynamics of the system is described in terms of coupled Langevin equations [25][26][27]: where γ R and γ φ are kinetic coefficients and the Gaussian white noises η and ξ have zero mean and correlations η α (t)η β (t ) = 2T R δ αβ δ(t − t ), (2.5a) ξ(r, t)ξ(r , t ) = 2T φ (−∇ 2 ) a δ(r − r )δ(t − t ).
The time-dependent statistics of R and φ is described by a probability distribution P (R, [φ], t) = P (R, R 0 , [φ], t, t 0 ), where R 0 ≡ R(t 0 ) denotes the tracer position at the initial time t 0 . Integrating P over the fluctuations of the OP field φ renders the marginal probability distribution P (R, t) ≡ Dφ P (R, [φ], t), (2.12) which, due to global conservation of probability, fulfills L 0 dRP (R, t) = 1. (2.13) One aim of the present study is to determine the time-dependence ofP (R, t) by deriving effective Fokker-Planck equations. Importantly, we assume the OP to always remain in thermal equilibrium. Accordingly, in the presence of a reactive tracer, the OP distribution P φ (R, [φ]) is time independent and given by P s in Eq. (2.1). In the passive case, instead, one has [see Eq. (2. 2)] which is independent of R. The initial joint probability distribution is given by . (2.15) Note that, at times t > t 0 , the joint distribution does not generally factorize into R-and φ-independent parts. For simplicity, we set t 0 = 0 and usually suppress the dependence of P on R 0 and t 0 . We will frequently express the tracer probability distributionP (R, t) in terms of the dimensionless coordinate ρ ≡ R/L, which formally renders a new distributionp(ρ, t) = LP (ρL, t). However, for notational simplicity we will use ρ also as a shorthand notation in the expressions forP .

A. Mode expansion and boundary conditions of the OP
We begin by introducing a set of eigenfunctions σ n (z) and eigenvalues k 2 n of the operator −∂ 2 z for the various BCs [see Eq. (2. 3)] considered in this study: The eigenfunctions are orthonormal: and complete: Note that complex conjugation is only relevant for periodic BCs. The eigenfunction expansions of the OP field and the noise take the form: For a real-valued function such as φ(z, t), one has φ −n (t) = φ * n (t). According to Eq. (2.5b), the correlations of the noise modes are given by We assume the initial condition φ(z, t = t i ) = 0 to apply in the infinite past (t i = −∞), such that at t = t 0 = 0 the OP is equilibrated [see Eq. (2.15)] and its initial condition plays no role anymore. The zero mode φ n=0 , which occurs (except for Dirichlet BCs) in the case of dissipative dynamics (a = 0), has to be treated separately. For all BCs except standard Dirichlet ones [Eq. (2.3b)], Eq. (2.10) with a = 1 conserves the OP globally, i.e., L 0 dz φ(z, t) = const. Standard Dirichlet BCs generally entail a non-zero flux through the boundaries, requiring to use suitable "no-flux" basis functions [60] instead in order to ensure global OP conservation. However, since this is technically involved, we do not consider conserved dynamics in conjunction with Dirichlet BCs in the following.
Due to the equilibrium assumption, the mean OP profile φ(z) is time-independent and, for the BCs specified in Eq. (3.1), in fact, vanishes, φ(z) = 0. A spatially inhomogeneous profile can arise here either due to a local bulk field (h = 0), representing a linearly coupled reactive tracer, or due to boundary fields (h 1 = 0). For a system with h = 0 and boundary fields of identical strength at z ∈ {0, L}, the mean OP profile is given by [see Appendix B 1] Capillary BCs are applicable only in the absence of a zero mode, as otherwise φ(z) h1 would be divergent [see Appendix B 1]. A system without a zero mode is naturally realized for conserved dynamics.

B. Solution of the OP dynamics
We next present exact as well as perturbative solutions of Eq. (3.7), which will be used in the course of this work.
1. Solution of Eq. For c = 0 and arbitrary ζ, as well as generally for ζ = 0 (passive tracer), the solution of Eq. (3.7) is given by: where τ n ≡ σ n (0) + σ n (L) and σ 0 = 1/ √ L. The zero mode φ 0 performs a diffusive motion superimposed on a linear growth. Since this growth depends on the initial time t i , the latter has to be kept finite in Eq. (3.10b), whereas in Eq. (3.10a), it is unproblematic to set t i = −∞ in the lower integration boundary. In the adiabatic limitχ 1, Eq. (3.10a) reduces, for ζ = 0 or h = 0, to [see Eq. (C2)] n , (n = 0,χ 1) (3.11) while the zero mode [Eq. (3.10b)] does not admit an expansion for smallχ. We conclude that BCs involving a zero mode are not compatible with the adiabatic limit. This is expected, since the relaxation time of a zero mode is infinite, which violates the assumption of a fast OP dynamics. Besides standard periodic or Neumann BCs, we shall thus also consider modified variants thereof for which the zero mode is explicitly removed [see Eq. (3.19) below]. Note that the adiabatic approximation can in general not be used to calculate equilibrium variances of φ (cf. Sec. III C 3).
2. Solution of Eq. For c = 0, a perturbative solution of Eq. (3.7) in orders of c can be constructed. To this end, we formally expand the OP as φ(z, t) = φ (0) + φ (1) + . . ., where φ (i) ∼ O(c i ), and assume the noise ξ ∼ O(c 0 ) [62]. Inserting this expansion in Eq. (3.7) (with h = 0) and grouping terms of the same order in c, we obtain n (t) is given by Eq. (3.10) with h = 0. Note that the only dependence on h 1 enters through φ (0) n . Determining, analogously to Appendix C, the adiabatic limitχ 1 of Eq. (3.12a), gives Order-parameter correlation functions The (connected) equilibrium OP correlation function is defined as usual as where δφ denotes the fluctuating part of φ. We provide in the following expressions for C φ within the Gaussian model and evaluate them in the adiabatic limit. Technical details are deferred to Appendix B.

Time-dependent correlation functions
The time-dependent correlation function follows straightforwardly from the solution of the Langevin equation [Eq. (3.7)] discussed above. In the case of a passive tracer (ζ = 0), Eqs. (3.6) and (3.10) render the equilibrium two-time correlator where the prime indicates a sum excluding n = 0 and we used Eqs. (3.8) and (B6) to identify the mean part φ(z) h1 , as required by Eq. (3.14). The last term in Eq. (3.15) stems from the zero mode and exists only for periodic or Neumann BCs [see Eq. (3.1)] and non-conserved dynamics (a = 0). In the following, we will also need the correlation function of ∂ z φ: 3)]. Since the zero mode does not contribute to C ∂φ , the latter is solely a function of the time difference. The correlation function C (p) for periodic BCs can be expressed in terms of the corresponding one for Neumann and Dirichlet BCs [63]: where the r.h.s. is to be evaluated for a system of size L/2 instead of L.

Static correlation functions
In equilibrium for t = t , the (connected) OP correlation function [see Eq. (B10)] can be directly determined from Eq. (2.7):   Here and in the following, ± refers to capillary BCs [Eq. (3.9)] [64], while a boundary condition labeled by an asterisk indicates that the zero mode is removed from the set of modes in Eq. (3.1). Physically, the vanishing of the zero mode can be ensured within conserved dynamics if a vanishing mean OP φ(t i ) = 0 is imposed as initial condition [see Eq. (3.7)]. However, in order to elucidate the effect of the conservation law, we will use (p * ) and (N * ) BCs also in conjunction with dissipative dynamics, if this is required to obtain a well-defined model (e.g., in the adiabatic limit, see Sec. III C 3 below). Note that, for capillary BCs, the OP correlation function takes a Neumann form (see Appendix B 1). As a characteristic of the Gaussian model, Eq. (3.19) holds independently from the presence of boundary or bulk fields (see Appendix B 1); in particular, it applies to an OP coupled linearly to a passive or reactive tracer. If the tracer is coupled quadratically, by contrast, the static correlation function differs in the passive and the reactive case (see Appendix B 2).
which turns out to be a central quantity for the tracer dynamics and is illustrated in Fig. 2(a). It is useful to remark that V φ (R) = φ (0) (R) 2 − φ 2 h1 , which follows from Eqs. (3.10) and (B6) (with ζ = 0). Note that we defined V φ only for BCs without a zero mode, as the variance is infinite otherwise.

Adiabatic approximation
The adiabatic limit of C φ can be obtained by inserting Eq. (3.11) into Eq. (3.14): and The corresponding expression for Neumann BCs with a = 1 is of similar polynomial form, but rather lengthy and not stated here. The expression for periodic BCs can be constructed by means of Eq. (3.18). Since the relaxation time of a zero mode is divergent [see also Eq. (3.15)], the actual limitχ,χ φ → 0 does not exist for standard periodic or Neumann BCs. We remark that i.e., the term on the right-hand side of Eq. (3.21) multiplying the δ-function corresponds to the time integral of the correlation function.  Using Eq. (3.21), one obtains for the expression of C ∂φ in the adiabatic limit. We introduce a function m(R) by writing For the various BCs, one has (see Fig. 2(b)): Note that m is dimensionless and we generally suppress the dependence on L. It turns out that m encodes the modification of the tracer mobility due to the OP fluctuations.

D. Effective Langevin equation for the tracer
The Langevin equation in Eq. (2.10a) can be rewritten in a more compact form: (3.27) wherein Ξ h and Ξ c represent effective forcing terms. In the case of a passive tracer, the only source of inhomogeneity for the OP are boundary fields h 1 = 0, which allows us to decompose the OP as φ(z) = φ(z) h1 + δφ(z), with the mean equilibrium profile φ(z) h1 stated in Eq. (3.8) and δφ(z) = 0. Accordingly, the mean parts of Ξ h and Ξ c give rise to (time-independent) effective potentials U h and U c : where V φ is the fluctuation variance stated in Eq. (3.20). It turns out that the effective potentials U h,c (see Fig. 3 We emphasize that Π c , which is essentially the square of the Gaussian process φ, is itself not Gaussian [65,66]. A Gaussian approximation can be made by assuming a small coupling c, such that higher-order cumulants of Π c become negligible. Due to the non-linear and non-Markovian nature of Eq. (3.27), its time-dependent solution can in general not be determined exactly. In the following, we thus solve Eq. (3.27) perturbatively, either by assuming a weak coupling or by applying an adiabatic approximation. The latter approach amounts to replacing the noises Π h and Π c by suitable Markovian approximations.

IV. PASSIVE TRACER
We begin by discussing a passive tracer, which is described by Eq. (2.10) with ζ = 0. Since the forcing term [first term on the r.h.s. of Eq. (2.10a)] is not balanced by a corresponding dissipation term in Eq. (2.10b), a passive tracer represents a driven, non-equilibrium system. We separately analyze the cases of a linear (h = 0, c = 0) and a quadratic coupling (h = 0, c = 0) between tracer and OP field.
A. Linear tracer-field coupling

Adiabatic approximation
We apply the adiabatic elimination procedure described in Refs. [67,68] in order to integrate out the fast dynamics of φ from Eq. (2.10), which results in the following FPE for the effective tracer probability distributionP (see Appendix E): where the prime denotes a derivative,χ, m(R), and U h (R) ∝ h 1 are stated in Eqs. (2.11), (3.26), and (3.28a), respectively, and we defined the effective dimensionless coupling constant The term D (R)/2 in Eq. (4.2a) represents a spurious drift [55], while the prefactor of U h (R) represents a mobility, which is unity here, i.e., in physical units [see Eq. (2.4a)], the effective and bare mobilities of a passive tracer are identical. As revealed by a comparison of the characteristic relaxation rates of the OP and the tracer, the adiabatic approximation requires not onlyχ 1, but, in fact,χκ h 1 [see Eq. (E10)]. The steady state solution of Eq. (4.1) (with vanishing flux at the boundaries) is given by [see also Eq. (D3)] The exponential term is absent for h 1 = 0. In this case, the normalization constant Z evaluates for a = 0 (non- κ h , and Z (N * ) = 2 arccot 2/ √χ κ h / √χ κ h ; for a = 1 and generally for h 1 = 0, Z has to be calculated numerically. We recall that standard Dirichlet BCs are not compatible with global probability conservation (a = 1). In the extreme adiabatic limitχ = 0, Eq. (4.4) reduces toP It is interesting to contrast the above results to a "naive" derivation of the adiabatic limit of Eq. (3.27): by using Eq. (3.11) in the expression for Π h in Eq. (3.29a), we obtain, to leading order in χ, a Gaussian Markovian (white) noise with correlations given by Eqs. (3.25) and (3.30a), i.e., which are consistent with Eq. (4.2b). However, Π h is also a multiplicative noise and thus requires specifying a stochastic integration rule [55,69]. The form of the spurious drift in Eq. (4.2a) suggests a S tratonovich convention, such that the Langevin equation associated with Eq. (4.1) takes the form (see Appendix D) where θ is a Gaussian white noise of zero mean.

Weak-coupling approximation
In a complementary approach to the adiabatic approximation [Sec. IV A 1], we determine here a perturbative solution in terms of h forP (R, t) [Eq. (2.12)]. As it turns out, one can thereby capture certain non-Markovian effects neglected in the adiabatic limit. To this end, we regard the first term on the r.h.s. of Eq. (2.10a) as a generic time-dependent force, which allows us to set up the following FPE for the "reduced" probability densityP (R, t) = P (R, R 0 , t, t 0 ) of the tracer [25,27,55]: with the fluxĴ Due to the reflective BCs for the tracer, one hasĴ R∈{0,L} = 0, (4.11) consistent with Eq. (2.13). Note that the reduced distributionP is distinct fromP in Eq. (2.12), since bothP andĴ depend implicitly on the OP field φ(z, t), which will be specified below in terms of its correlations [see Sec. III C] [70]. Averaging over the OP fluctuations described by Eq. (2.14), relevant for the passive case, yields an approximation for P :P We now take c = 0 and h 1 = 0 and formally decompose the reduced distribution asP =P 0 +P 1 +P 2 +. . ., assuminĝ , which can be made explicit by appropriate rescaling.) Due to vanishing boundary fields, we have φ = 0. Accordingly, since Eq. (4.9) (with c = 0) is linear in φ, the first non-trivial influence of φ arises at O(h 2 ). Inserting the expansion ofP into Eq. (4.9) and collecting terms of the same order in h, renders The solution of Eq. (4.14) SinceP =P 0 [see Eq. (4.12)] already validates Eq. (2.13) at all t, we require The solution of the inhomogeneous Fokker-Planck equation in Eq. (4.13b) can be stated in terms of the associated Green's function G (determined below) aŝ We assume no-flux BCs [Eq. (4.11)] to apply also toP 1 . Together with Eq. (4.15), this implies that the Green's function takes the same form asP 0 in Eq. (4.14) except for the absence of the term with n = 0: We remark that the completeness relation in Eq.
The solution of Eq. (4.13c) follows in an analogous way: Upon averaging over the fluctuations of φ [see Eq. (4.12)], we obtain the distributions with C ∂φ defined in Eq. (3.16). By means of Eq. (3.18), the expression of Eq. (4.20b) for periodic BCs on φ can be obtained from the corresponding expressions for Neumann and Dirichlet BCs as In order to evaluate Eq. (4.20b), we use Eqs. (4.14) and (4.17) and subsequently perform the spatial and temporal integrations. For Neumann BCs, the resulting expression is convergent and simplifies to three infinite sums over the mode indices, which can be efficiently evaluated numerically owing to the exponential term. For Dirichlet BCs, instead, one is left with four infinite sums [which can be reduced to three sums in the late-time limit, see Eq. (4.26)] andP 2 grows logarithmically with the summation cutoff. This divergence appears to be an artifact of the perturbative approach, as it turns out that the shape ofP still qualitatively describes the numerical results (see Sec. VI). For simplicity, we henceforth set t 0 = 0 [see Eq. (2.15)]. In order to determine the late time limit ofP 2 , we reorder the time integration variables in Eq. (4.20b) to obtain  [71], where k min is the minimal wavenumber in the system and we recall that a = 0 (a = 1) for dissipative (conserved) OP dynamics. When considering the limit t → ∞, the integrand thus gives substantial contributions only for v t and w t − v. Accordingly, we may replace P 0 [Eq. (4.14)] by its late-time limit,P which is independent of position. This allows us to extend the upper limit of the w integral from t − v to t, resulting inP The time and space integrals can be performed explicitly, rendering, for Neumann BCs on the OP: with the effective dimensionless coupling constant κ h defined in Eq. (4.3). The weak-coupling approximation applies to the regime κ h 1. For Dirichlet BCs (which are relevant only for a = 0), instead, we obtain , (4.26) where the function J n arises from the spatial integrals and is given by with ρ ≡ R/L andχ given in Eq. (2.11). The numbers β l (N ) have to be determined numerically from Eq. (4.26) and are found to decay exponentially with increasing l (for large l) and grow logarithmically with N . The corresponding distribution for periodic BCs can be obtained via Eq. (4.21). We remark that, forχ → 0, the distributions in Eq. (4.28) become uniform,P s → 1/L. This can alternatively be shown by inserting the adiabatic approximation for C ∂φ reported in Eq. (3.24) into Eq. (4.24). Here and in the following, we use a notation such as (D±D) to indicate a tracer linearly coupled to a OP field obeying Dirichlet BCs. For an OP subject to Neumann BCs, both the adiabatic and the weak-coupling approximation yield an increased occupation probability at the boundaries and a reduction at the center of the system. For Dirichlet BCs, in contrast, the two approaches predict opposite behaviors. Interestingly, numerical simulations indicate that both approximations describe certain aspects of the actual tracer distribution (see Sec. VI for further discussion). In particular, the attraction of the tracer towards the wall is generically expected for a confined non-Markovian process. In the case of capillary BCs, the steady-state distribution [see Eq.  Figure 5 illustrates the time evolution of the tracer distributionP (R, t) in the adiabatic limit, as obtained from Eq. (4.1) for Neumann BCs of the OP. The time evolution ofP is found to be slightly faster for dissipative (a) than for conserved dynamics (b). The dissipative dynamics for Dirichlet BCs follows analogously, but does not exhibit any new features and is thus not shown.

Discussion
If periodic BCs are imposed on the OP, Eq. (4.2) reduces to µ = 0 and D = T R , such that Eq. (4.1) describes a simple diffusion process subject to reflective BCs, which hasP s = 1/L as steady-state solution. However, numerical simulations (not shown) reveal this to be an artifact of the present order of the adiabatic approximation. In fact, within the weak-coupling approximation, Eq. (4.21) implies that the behavior of the tracer is similar for periodic, Neumann or Dirichlet BCs. The adiabatic and the weak-coupling approximations need not necessarily agree in the limitχ 1, because, in the former approach the field φ is interpreted as a noise, whereas in the latter, it is formally regarded as a time-dependent potential.
B. Quadratic tracer-field coupling

Adiabatic approximation
In the case of a quadratically coupled passive tracer, the Langevin equation in Eq. (3.27) (with h = 0) reduces tȯ where the effective potential U c and the noise Π c are reported in Eqs. (3.28b) and (3.29b), respectively. We apply the adiabatic elimination procedure described in Refs. [55,72] in order to obtain an effective FPE for the tracer distributionP (R, t). We furthermore assume the coupling c to be small, such that Π c can be approximated as a Gaussian Markovian white noise with correlation (see Appendix F) where the amplitude P ∼ O(c 2 χ) is specified in Eq. (F3). Within these approximations, the FPE associated with Eq. (4.29) follows as (see §4.8 in Ref. [72]) with the effective drift and diffusion coefficients where we expanded the potential V up to O(cχ 0 ) and disregarded any R-independent constants. SpecializingP s to the various BCs results in (ρ ≡ R/L) with the dimensionless effective coupling The above assumption of a weak coupling implies thatP s in Eq.

Weak-coupling approximation
Here, analogously to Sec. IV A 2, we develop a perturbative solution of the tracer dynamics by assuming a weak coupling c [the actual dimensionless control parameter being κ c , see Eq. (4.35)]. We thus consider Eq. (4.9) with h = 0 and formally expand the reduced tracer distribution asP =P 0 +P 1 +P 2 + . . ., withP i ∼ O(c i ). Upon inserting this expansion into Eq. (4.9), one obtains the hierarchy The solution forP 0 is given in Eq. (4.14). For the leading correction, we obtain, analogously to Eq. (4.16): While Eq. (4.38) has to be evaluated numerically in the general case, the steady-state distributionP s up to O(c) can be readily determined from Eqs. (4.36a) and (4.36b): the former yieldsP s,0 = 1/L, whereas the latter reduces, after averaging over the field, to [see Eq. (3.28b)] Upon imposing the no-flux condition [Eq. (4.11)], we obtain where the integration constant α is fixed via Eq. (4.15) and we introduced the dimensionless coupling These expressions agree with the ones obtained from the long-time limit of Eq. (4.38a) (not shown) and from the expansion of Eq. (4.34) to O(κ c ). A dependence ofP s on χ and a, i.e., the conservation law [see Eq.

Discussion
If non-symmetry-breaking BCs are imposed on the OP, in both the adiabatic [Eq. (4.31)] and the weak-coupling approximation [Eq. (4.36b)] the tracer follows, to leading order in κ c andχ, a Brownian motion in a quadratic potential U c (R) subject to reflective BCs [see Eqs. (3.20) and (3.27) and Fig. 2(a)]. For Neumann BCs, this is, in fact, a confined Ornstein-Uhlenbeck process. While the time-dependent solution for such processes can in principle be determined analytically [73], the resulting expression is rather involved and we instead present in Fig. 6 a numerical solution of Eq. (4.31). Since the behavior of (NcN) and (DcD) is qualitatively similar, only the latter case is shown in the figure.
According to Eqs. For capillary BCs, the tracer behaves to leading order as a Brownian particle in an effective potential U c [see Eq. (3.28b) and Fig. 3]. The latter is a fourth-order polynomial and shows a crossover from a unimodal shape (|H 1 | 1) with a minimum at the center of the system, to a bimodal shape (|H 1 | 1) with minima at

V. REACTIVE TRACER
We now turn to the discussion of a reactive tracer, which is described by Eq. (2.10) with ζ = 1. Since Eq. (2.10) satisfies in this case a fluctuation-dissipation relation for all degrees of freedom [40,41,74], the resulting joint steady-state distribution for φ and R is provided by Eqs. (2.1) and (2.2). A reactive tracer represents a simplified model for a colloid in a fluid. While previous studies of the equilibrium behavior of a single confined colloid considered spatially resolved particles in a half-space [11][12][13][14][15][16] or in a slit geometry [75][76][77], we focus here on a point-like particle under strong confinement. The confinement induces long-ranged interactions between the colloid and both boundaries (walls). Further connections to previous studies are discussed in Sec. VI B 2.
In Sec. V A, we determine the equilibrium distribution of the tracer for various couplings and OP BCs. Besides being interesting in its own right, the equilibrium distributions provide a means to independently check the steady-state solutions obtained from the effective FPEs derived in Sec. V B. While we focus here on a one-dimensional system, equilibrium distributions of a tracer in a three-dimensional slit are presented in Appendix H.

A. Equilibrium tracer distribution
The equilibrium (steady-state) probability distribution for the tracer follows by marginalizing the joint distribution in Eq. (2.7) over the OP field φ [see Eq. (2.12)]: Owing to the Gaussian nature of Eq. (5.1), we can integrate out the field degrees-of-freedom by inserting the orthonormal transformation in Eq. (3.4), which induces a unit Jacobian. The actual integration is performed over the modes φ n with the aid of the following standard result for multidimensional Gaussian integration [78,79]: where Γ is a N × N matrix and K i is a given field. Complex conjugation is relevant only for periodic BCs, in which case the modes φ n and K n are complex-valued with φ −n = φ * n (analogously for K n ) and the integration measure has to be suitably chosen [78,80]. A possible zero mode, occurring for periodic and Neumann BCs, can be regularized by replacing the vanishing eigenvalue k 0 [see Eq. (3.1)] by a nonzero parameter ε [81], which is set to zero in the end of the calculation [see also Appendix G].

System without boundary fields
We consider first a system without boundary fields, i.e., h 1 = 0. In the case of a linear coupling between OP and tracer (i.e., h = 0, c = 0), Eq. (5.1) evaluates tō 3) reduces toP s (R) 1/L, as expected. In the limit κ h → ∞, instead, using the asymptotic behavior erfi(x → ∞) exp(x 2 )/ √ πx, the tracer becomes highly localized for Neumann or Dirichlet BCs on the OP:  Fig. 7(a). If the OP is subject to Dirichlet BCs, the probability of the tracer is enhanced at the center of the system and reduced at the boundaries, while the trends are opposite in the case of Neumann BCs. These behaviors are qualitatively similar to a passive tracer in the adiabatic limit (see Fig. 4(a)). This can be intuitively explained based on the dynamical coupling ∂ R φ(R) [see Eq. (2.10a)], which acts as an effective noise [see Eq. (3.27)] in the Langevin equation: for Neumann (Dirichlet) BCs, this noise is strongest at the center (boundaries) and thus drives the tracer towards the boundaries (center). Note, however, that this effect can be overwhelmed by non-Markovian contributions, responsible for the generally strong attraction of a passive tracer towards the boundaries (see discussion in Sec. VI).

Equation (5.3) is illustrated in
For a tracer quadratically coupled to the OP (i.e., c = 0, h = 0), Eq. (5.1) evaluates tō where the matrix Γ is given by [82] and the continuum limit of its determinant is calculated in Appendix G. Upon normalization, Eq. (5.5) results in 1 with the dimensionless coupling constant κ c in Eq. (4.35). In the limit κ c → ∞, the tracer effectively imposes Dirichlet BCs on the OP field, such that Eq. (5.7), becoming independent of κ c , reduces tō 1 arcosh Accordingly, if the OP is subject to Dirichlet (Neumann) BCs, a quadratically coupled tracer is most likely to be found at the boundaries (center) of the system, as illustrated in Fig. 7(b). This behavior is similar to the passive case [see Eq.

System with boundary fields
We assume now boundary fields of equal strength h 1 to act on the OP. Upon performing in Eq. (5.1) the Gaussian integration over φ and omitting all R-independent constants, one obtains the following equilibrium distribution for a linearly coupled tracer (c = 0): reported in Eqs. (3.19) and (3.20), respectively. Note that the term in the last square brackets essentially simplifies to φ(R) h1 [see Eq. (B11)], such that this part of the distribution resembles Eq. (4.5) of the passive case. The normalized tracer probability distribution is given by (with ρ ≡ R/L) with κ h1 ≡ T φ L|h|(2|h 1 | − |h|). This distribution has its maximum at L/2, indicating a repulsion between wall and tracer [see Fig. 8(a)]. In fact, in the limit κ h1 → ∞, the tracer is strongly localized at the center,P s (R) δ(R − L/2).
If the two boundary fields have opposite signs, we obtain asymmetric distributions, which, for sufficiently large |h 1 |, are monotonous and have a maximum at one boundary.
In the case of a quadratically coupled tracer with non-vanishing boundary fields (i.e., h = 0, h 1 = 0), Eq. (5.1) yieldsP . Taking, as required in this case, σ n to be Neumann modes without a zero mode, one obtains where the dimensionless couplings κ c and H 1 are reported in Eqs. (4.35) and (4.41), respectively, and the normalization factor Z =Z/L has to be calculated numerically. If a zero mode is present, one obtains a flat distribution,P s (R)| h=0 = 1/L, instead [see Eq. (G11)]. In the limit κ c → ∞, the OP obeys Dirichlet BCs at the tracer location and the distribution in Eq. (5.14) becomes independent of κ c , reducing tō Upon increasing the parameter H 1 , the distributions in Eqs. (5.14) and (5.15) show a cross-over [see Fig. 8(b)] from a single-to a double-peaked shape with peaks located at In the limit H 1 → ∞, Eqs. (4.42) and (5.16) coincide andP s reduces to a sum of two Dirac-δ functions at R ± .

Many tracers
The above results can be straightforwardly generalized to N p > 1 tracers in the system. To this end, the terms in the square brackets in Eq. (5.1) are replaced according to hφ(R) → Np j=1 h j φ(R j ) and cφ 2 (R) → Np where h j and c j denote the coupling constants pertaining to the jth tracer. As before, boundary fields of equal strengthh 1 = h 1 /T φ act on the OP φ. In the case of a linear coupling between each tracer and the OP, Eq. (5.9) generalizes toP withh k ≡ h k /T R . In the final result, we omitted all terms independent of R k , as they are canceled by the normalization.
While the sums in Eq. (5.17) can be calculated for any number of tracers, we focus in the following on N p = 2 and couplings of equal magnitude |h j | = h. In the case of Dirichlet BCs on φ, the contributions associated with the boundary field h 1 identically vanish and one is left with with ρ k ≡ R k /L,h = h/T R , and a normalization factor Z that can be calculated analytically. In the case of capillary BCs on φ (which, as before, requires taking the σ n to be Neumann modes), the joint probability distribution of the two tracers is given bȳ (++−+) (5.19) When the OP obeys Neumann BCs, the distribution follows by setting h 1 = 0 in this expression. Figure 9 illustrates the probability distribution of two linearly coupled reactive tracers in confinement. The essential features can be readily understood based on the behavior of a single tracer [see Figs. 7(a) and 8(a)]. To this end, we first note that two tracers having couplings of the same (opposite) sign attract (repel) each other. This behavior directly reflects the interactions of a single tracer with a boundary field [see Fig. 8(a)] and is well-known in the context of critical Casimir forces (see Sec. VI B 2). Accordingly, two tracers with (++)-type couplings tend to occupy the same region in the system, which, in the case of Dirichlet BCs, is in the center [ Fig. 9(a)], or, in the case of Neumann or capillary BCs of the same sign, at the boundaries [ Fig. 9(c,e)]. For the latter two BCs, due to the attractive interactions between tracer and boundary, the tracers also have a nonzero probability to be located at opposite walls [in the case (N++N), the effect is suppressed except for small h, and is hence not visible in Fig. 9(c)]. By contrast, two tracers coupled linearly with opposite signs [(+−)] to an OP subject to Neumann BCs likely reside at different boundaries of the system [ Fig. 9(d)]. If the interactions between boundary and tracer are (partly) repulsive [as for (D±), see Fig. 7(a)], one tracer tends to be located near the center of the system, while the other resides at a boundary [see Fig. 9(b,f)]. The above calculations can be readily extended to tracers with quadratic or mixed couplings.

B. Dynamics
We discuss in the following the effective dynamics of a reactive tracer in the adiabatic regime, i.e., assuming the OP field φ to be a fast variable. This regime is typically realized in experiments on colloidal particles in critical solvents [24].

Linear tracer-field coupling
We first consider a tracer linearly coupled to a critical fluctuating OP field. This situation approximately describes a colloidal particle subject to critical adsorption. In order to perform the adiabatic elimination of φ in the Langevin equations in Eq. (2.10), we apply the method of Refs. [67,68], which renders the following FPE for the effective tracer distributionP (see Appendix E): with the drift and diffusion coefficients given by [see Eq. (E3)] is an effective potential [see Eqs.
where θ is a Gaussian white noise. We remark that, using any other convention for the noise requires adding a spurious drift term to the Langevin equation in Eq. (5.24) in order to recover the correct form of the drift in Eq. (5.21a) (see Appendix D). It has been previously noted that the isothermal convention is indeed a natural choice for the Langevin description of particles in an equilibrium system [83,86]. Here, we have rigorously derived the underlying FPE from a system of stochastic differential equations with additive noise [Eq. (2.10)], for which there is no ambiguity in its interpretation.

Quadratic tracer-field coupling
We now determine the effective dynamics of a point-like colloidal particle quadratically coupled to a critical medium. In the strong coupling limit, c → ∞, the particle imposes Dirichlet BCs on the OP at R. However, in the present approach we are concerned with the opposite limit of a weak coupling. We focus on the dynamics to O(c), which is obtained (analogously to the passive case, see Sec. IV B 1) by inserting the adiabatic weak-coupling solutions for φ reported in Eq. (3.13) into the Langevin equation in Eq. (3.27). Applying the adiabatic elimination procedure of Refs. [55,72] [87], using the O(c 0 )-expression φ (0) (R) 2 = V φ (R) + φ(R) 2 h1 = (2/c)U c (R) [see Eqs. (3.8) and (3.20)], renders the linear Langevin equationṘ (t) = −∂ R U c (R(t)) + η(t), (5.25) with the potential U c reported in Eq. (3.28b). The associated FPE is given by (5.26) which coincides with the one for a passive tracer [see Eq. (4.32)] at O(c). Accordingly, the associated steady-state distribution is given by Eq. (4.33), i.e.,P Numerical simulations (see Sec. VI) indeed confirm that, for a quadratically coupled tracer, steady-state distributions in the passive and reactive cases are similar [88].

VI. SIMULATION RESULTS & DISCUSSION
In the next subsection, we present results for the stationary distribution and the mean squared displacement of the tracer obtained from numerical Langevin simulations of Eq. (2.10). Furthermore, we discuss the generic mechanism underlying the observed behaviors (Sec. VI B 1) and place our results in the general context of boundary critical phenomena (Sec. VI B 2).

A. Simulations
Numerical results for the tracer position are generated by solving the Langevin Eq. (2.10a) in real space (for a system size of L = 100 in simulation units) and Eq. (2.10b) in mode space using Eq. (3.7) (with a total number of 100 modes) [89]. For both equations a standard Euler forward integration scheme is employed [55] Fig. 12). We remark that, since standard Dirichlet BCs violate global OP conservation (see discussion in Sec. III A), we consider in this case only non-conserved dynamics. For all other BCs, we typically study both conserved as well as non-conserved dynamics. Simulation data is recorded after an initial transient period [of approximate duration χ(L/π) 2+2a ] required to equilibrate the OP field.

Statics
In Fig. 10, the numerically determined steady-state distributionsP s (R) of a tracer are summarized, focusing on OP fields obeying Dirichlet or Neumann BCs. A passive tracer (first row) is genuinely out of thermal equilibrium and thusP s depends, in principle, on the specific dynamics and the conservation law. However, for Neumann BCs, simulations reveal only a marginal difference between conserved (dashed curves) and dissipative dynamics (thin curves with filling) in the considered parameter regime. Since, for a quadratically coupled tracer, the presence of a zero mode would lead to a uniform distributionP s = 1/L [see Eq. (5.7)], we consider in that case only Neumann BCs without a zero mode (N * cN * , last column). While this is easily arranged in a simulation with dissipative dynamics, it requires conserved dynamics in an actual physical setup.
In the case of a reactive tracer (second row), the equilibrium distributions in Eqs. The local maximum ofP s at R = L/2 observed in Fig. 10 in the case of a passive tracer for (D±D) is predicted by the adiabatic approximation [see Eq. (4.4) and Fig. 4(a)]. This is detailed in Fig. 11(a), which showsP s determined from simulations in the adiabatic regime for (D±D), non-conserved OP dynamics, and various values ofχκ h . As the latter parameter is decreased, the central maximum ofP s grows at the expense of the maxima at the boundaries. Figure 11(b) illustrates the corresponding behavior ofP s in the case (N±N) for conserved and non-conserved OP For all other couplings and BCs, one obtains two Dirac-δ-like distributions located at the boundaries (represented by thin vertical bars in the plot). This behavior can be readily understood by noting that, in Eq. (3.27) the coupling terms ∂ R φ and φ vanish at the boundaries for Neumann and Dirichlet BCs, respectively. Interestingly, in the case (N * cN * ), the transition from T R = T φ to T R = 0 proceeds by a growth of the central peak ofP s and a rather sharp increase of the probability at the boundaries. Figure 12 illustrates the steady-state distribution of a tracer in the presence of non-vanishing boundary fields h 1 . In the passive case (top row),P s is dominated by the deterministic potentials U h and U c [see Eqs. (4.5) and (4.34)] and is thus inhomogeneous even forχ = 0, in contrast to the case with non-symmetry-breaking BCs (h 1 = 0). We find that, forχ 1, the simulation results (thin curves with filling) are accurately captured by the expressions obtained based on the adiabatic approximation in Eqs. (4.4) and (4.34) (thick curves). As was the case for h 1 = 0, the analytically determined equilibrium distributions for a reactive tracer in Eqs. (5.10), (5.12), and (5.14) (thick curves, bottom row) exactly match the numerical results.

Dynamics
In order to assess the tracer dynamics, we consider the mean-squared displacement (MSD) of the tracer location, where we take R 0 ≡ R(0) = L/2 as initial position. Figure 13 illustrates the MSD of a tracer coupled linearly to a (non-conserved) OP field with Dirichlet BCs in the adiabatic regime. In simulation, the average in Eq. (6.1) is obtained over multiple stochastic realizations of the noise, whereas the theoretical prediction is determined by inserting in Eq. (6.1) forP the (numerically calculated) solution of the FPEs reported in Eqs. (4.1) and (5.20). Since these FPEs describe a Markovian process with time-independent drift and diffusion coefficients, the resulting MSD with the diffusivity D(R) given in Eqs. (4.2b) and (5.21b). For a passive tracer within the adiabatic regime, the OP acts as an additional Markovian noise source [cf. Eq. (3.27)], causing the diffusivity to surpass the free one T R . In contrast, a reactive tracer polarizes the surrounding medium, which hinders displacement and consequently reduces the diffusivity relative to the free one. These behaviors are consistent with the results of Ref. [25] and we emphasize that the trends observed when varyingχ apply only toχ 1. Remarkably, already forχκ h O(10), our simulation results (solid curves) are accurately captured by the analytical predictions (dashed curves). At shorter times as well as for largerχκ h , the non-Markovian character of the OP fluctuations becomes prominent, causing the simulations to increasingly deviate from the adiabatic approximation [see Eq. (E10)]. In fact, for t → 0, the MSD obtained from the simulations approaches the one of a free (h = 0) tracer, ∆R(t t R ) 2 = 2T R t [dashed-dotted curve, see inset in Fig. 13(a)]. We remark that, even in the Markovian regime, the spatially heterogeneous character of the diffusivity D(R) can lead to non-Brownian diffusion [83,92,93]. A more detailed analysis of the diffusivity will be performed in a separate study.
B. Discussion

Passive tracer
The steady-state distributions of a passive tracer (first row in Figs. 10 and 12) can be understood by inspecting the forcing terms Ξ h,c in the Langevin equation in Eq. (3.27). In the presence of boundary fields h 1 = 0, the tracer dynamics is essentially controlled by the non-vanishing mean OP profile φ(z) , which gives rise to the deterministic potentials U c,h (z) [see Eqs.  Fig. 14 as a function of R and ∆t around some fixed location R 0 near one boundary [94]. Note first that C ∂φ (R, R , ∆t) is positive if R is near R . Assume now that the tracer is located near a boundary and receives a "kick" from the noise which, in the absence of a boundary, would move the tracer beyond it. Due to the boundary condition, however, the tracer is reflected back to a position, where, owing to the positive temporal correlation of the noise, it is likely to again be kicked towards the boundary in the next time step. As a consequence, a passive tracer linearly coupled to a uniform OP field has an enhanced probability to reside near a boundary, as observed in Fig. 10. This effect is, in fact, generically expected for a confined stochastic process driven by a temporally correlated noise [95,96]. The dynamics of a quadratically coupled tracer, by contrast, is dominated by the non-vanishing mean of the forcing term Ξ c [Eq. (3.28b)], as described by the deterministic potential U c (R) [see Fig. 3 and Eqs. (4.33) and (5.27)].

Reactive tracer
The equilibrium distributionP s of a reactive tracer (second row in Fig. 10 and Fig. 12) encodes the fluctuationinduced interactions between inclusions in a critical medium [2,3,46]. Accordingly, it is informative to connect the present results to previous studies of the critical Casimir force (CCF) acting on a spatially extended spherical particle in front of a planar wall [11,14,15]. To this end, the Hamiltonian coupling of the point-like tracer [see Eq. (2.2)] must be mapped to a boundary condition for the OP φ at the particle surface: a linear coupling (h = 0, c = 0) corresponds to a + (or −) boundary condition, while a quadratic coupling (c = 0, h = 0) corresponds to a Dirichlet boundary condition (provided that |h| and |c| are sufficiently large). In the limit where the particle-wall distance R is large compared to the particle radius , a small-sphere expansion renders the Casimir (excess) free energy [11,14,15] Here, ψ = φ and x φ = β/ν, if both the wall and the particle impose symmetry-breaking BCs [i.e., (±±)], whereas ψ = φ 2 , x φ 2 = d − ν −1 , if neither particle nor wall, or only one, impose symmetry-breaking BCs [corresponding to the cases (±, D/N), (D,D/N), (NN)]. (Here, β and ν denote the standard bulk critical exponents.) The amplitudes A ψ a and B ψ and the exponent x ψ are defined via the associated bulk correlation function, ψ(r)ψ(r ) bulk = B ψ r −2x ψ , and the profile in the half-space with boundary condition a, ψ(r ⊥ ) a half-space = A ψ a (2r ⊥ ) −x ψ . The excess free energy contributes to the Casimir potential U(R) = F cas (R)+U add (R), which enters a Boltzmann-like probability distribution for the tracer [6],P The potential U add (R) accounts for additional interactions, such as those stemming from van der Waals forces, which are relevant at short distances R and regularize a possible divergence of F cas for R → 0 [13,97]. We do not consider these here. The CCF acting between particle and wall follows from Eq. (6.3) as K = −dF cas /dR = x ψ F cas /R. Since F cas in Eq. (6.3) is singular for a point-like particle ( → 0) and, moreover, does not include the effect of the second, distant wall, we do not expected quantitative agreement with the results obtained in our study. We thus focus instead on the sign of the CCF and asymptotic behavior of F cas : for the BCs considered here, Eq. (6.3) implies an attractive CCF for the combinations (a, b) = (D,D), (N,+), (+,+), and a repulsive one for (D,N), (D,+), (+,−) (see also Ref. [15]). These predictions are consistent with the behaviors observed in Figs. 10 and 12. For the Gaussian model in d = 1 dimensions, the exponent x ψ takes the value x φ 2 = −1 if either particle or wall (or both) have non-symmetry breaking BCs. This implies that F cas ∝ R as R → 0, in agreement with the asymptotic behavior resulting from (the negative logarithm of) the expressions in Eqs. (5.3), (5.7), and (5.14) [98]. In d = 3 dimensions (see Appendix H), we have x φ = x φ 2 = 1 within the Gaussian model, which agrees in the cases (D/N, ±,D/N) and (±±±) with the asymptotics reported in Eq. (H11).

VII. SUMMARY
We have investigated in this study the behavior of a confined point-like tracer particle coupled to a fluctuating order parameter (OP) field φ within the Gaussian approximation. The OP field represents a critical fluid medium in equilibrium and follows either dissipative or conservative dynamics (model A/B [39]). The tracer is governed by a Langevin equation [see Eq. (2.8)] and is subject to reflective BCs. We have considered passive as well as reactive types of tracers. The former is out of equilibrium, since the coupling to the fluctuating fluid represents an energy source that is unbalanced by dissipation. In this sense, a passive tracer bears resemblance to an "active" particle [43,44,57,99]. By contrast, a reactive tracer interacts with the fluid in accordance with the fluctuation-dissipation theorem, such that its steady state obeys equilibrium statistical mechanics. A reactive tracer can be viewed as a simplified model of a colloidal particle in a critical fluid. The action of a reactive tracer on the fluid is described either by a local chemical potential (linear coupling) or a locally altered correlation length (quadratic coupling) [see Eq. (2.2)]. A linear coupling enhances the OP around the tracer, as is typically observed for colloids [13,100]. We have also considered non-vanishing boundary fields (h 1 = 0). These induce a non-uniform OP profile, which manifests as a deterministic force acting on the tracer [see Eq. (3.28)].
The central quantity in our study is the probability distributionP (R, t) of the tracer position R. While previous studies of tracers in fluctuating media focused mostly on bulk systems [25][26][27], we have addressed here the effect of spatial confinement by considering the fluctuation-induced interactions of the tracer with two fixed boundaries (at R = 0 and L). This complements investigations of the critical Casimir force for (spatially extended) colloidal particles in a half-space [11,14,15] and in strong confinement [75][76][77]. In order to make analytical progress, we have focused on d = 1 spatial dimensions and employed adiabatic as well as weak-coupling approximations. In the adiabatic regime, the dynamics of the OP is fast compared to the one of the tracer, such that the effect of the OP can be approximated as a (spatially correlated) Markovian noise. This enables a description of the tracer dynamics in terms of a Fokker-Planck equation with (spatially dependent) drift and diffusion coefficients. In the case of a (linearly coupled) passive tracer, the nonlinear multiplicative noise in the associated Langevin equation can be interpreted in the Stratonovich sense, whereas, in the reactive case, an "isothermal" interpretation (also called anti-Ito or Hänggi-Klimontovich prescription [83][84][85]) emerges naturally. Note that the adiabatic approximation applies only to systems which do not involve a zero mode, such that the otherwise diverging relaxation time of a critical fluid [41] is cut off. One of our main results is given by Eq. (5.20), which describes the effective (adiabatic) equilibrium dynamics of a point-like colloidal particle in a confined critical medium in the presence of critical adsorption, i.e., a local enhancement of the OP around the particle.
We have validated our analytical results by numerically solving the associated Langevin equations [Eq. (2.10)] to obtain the steady-state distribution [see Figs. 10 to 12] as well as the mean-squared displacement [see Fig. 13]. While we focused a one-dimensional system, analytical calculations of the equilibrium distribution of a reactive tracer in three dimensions revel that they are qualitatively similar to the one-dimensional case (see Appendix H). A reactive tracer typically obeys a Boltzmann-like equilibrium distribution,P s (R) ∼ exp(−U(R)/T ), with an effective potential U(R). The latter is a consequence of the coupling to the OP field [see Sec. The effective potential U determines also the dynamics of the tracer at the leading order in the adiabatic and the weak-coupling approximations [see Eqs. (4.1), (4.31), (5.20), and (5.26)]. Since, within the Gaussian model considered here, δφ(R) 2 and φ(R) are quadratic functions of R, the tracer can in certain cases be effectively described by a confined Ornstein-Uhlenbeck process [101]. Beyond leading order, deviations from this simple behavior arise because the mobility acquires a spatial dependence [see discussion after Eqs. (4.1) and (5.20)]. Note that the adiabatic dynamics of a quadratically coupled reactive tracer has been considered here only to O(c) and the inclusion of higherorder corrections is reserved for a future study. In the adiabatic regime, the mean-squared displacement of the tracer grows linearly in time, with an effective diffusivity that increases (decreases) with the adiabaticity parameterχκ h [see Eq. (E10)] in the passive (reactive) case (see Fig. 13).
A linearly coupled passive tracer has a higher probability to be located near the boundaries than in the center of the system. This is a generic effect resulting from the interplay between confinement and a temporally correlated noise [95,96]. It also arises in the case of active matter, where it gives rise to the accumulation of active particles at surfaces and plays a role for motility-induced phase separation [43,44,99]. The behavior of a quadratically coupled (passive or reactive) tracer, instead, is dominated by the effective potential ∝ c φ(R) 2 , stemming from the nonzero average of the OP-related noise ∼ φ 2 [see Eq. (3.27) and Fig. 2(a)]. Interestingly, in the parameter regimes considered here, the steady-state distributions of a passive and a reactive tracer are qualitatively similar (see Figs. 10 and 12). An exception is a tracer coupled linearly to an OP field subject to Dirichlet BCs (see Fig. 10), in which case the steady-state distribution exhibits a crossover behavior controlled by the non-Markovian character of the dynamics [see Fig. 11(a)]. We finally remark that the steady-state distribution of a passive tracer is similar for dissipative and conservative OP dynamics. The present study opens up various possibilities to investigate colloidal dynamics in a critical medium within an analytical approach. In particular, our model can be readily extended to more than one tracer (see Sec. V A 3), which could be utilized to address many-body critical Casimir interactions [75,[102][103][104][105][106]. Furthermore, while we assumed the OP to remain in equilibrium at all times, non-equilibrium scenarios such as OP quenches [33,63,107,108] appear to be a rewarding topic. It is also pertinent to extend the present work towards two and three spatial dimensions, which are the relevant cases for membranes [28-30, 109, 110], interfaces [111][112][113], and colloidal suspensions [6]. Moreover, effects of off-criticality as well as hydrodynamics [114,115] could be taken into account in the future. The Gaussian and weak-coupling approximations employed here provide the leading order contributions of a perturbation expansion of a φ 4 -theory [46]. In fact, the couplings are expected to flow under a renormalization group and attain fixed-point values which, depending on the dimension, are not necessarily small. This should be addressed in a future study. Further attention should also be devoted to non-Markovian effects in the dynamics, which become relevant at large coupling strengths.

Appendix A: Dimensional considerations
From the fact that the argument of the exponential in Eq. (2.7) must be dimensionless, one infers the following dimensions of the field and the static couplings:

System with a linearly coupled tracer or boundary fields
We consider an OP field φ, which may be subject to boundary fields h 1 as well as to a bulk field h (at location R, representing a linearly coupled tracer). Connected correlation functions of φ can be determined in the usual fashion [79] from the generating functional which is obtained by introducing a position-dependent auxiliary field J(z) into the Hamiltonian in Eq. (2.2): We first consider the averaged profile Switching to mode space, we write J(z) = n σ n (z)J n and use Eqs. (3.2) and (3.4) to bring the Hamiltonian into the formĤ withτ n ≡ h 1 [σ n (0) + σ n (L)] + hσ n (R). In order to regularize a possible zero mode k 0 = 0, we set k 0 = > 0 and perform the limit → 0 at the end of the calculation. After a Gaussian integration using Eq. (5.2), the generating functional takes the form [116] Z where we omitted an unimportant normalization factor. The mode-space expression for the averaged profile resulting from Eq. (B3) follows as The zero mode k 0 = → 0 renders a divergent mean profile in equilibrium if h = 0 or h 1 = 0. In the absence of a zero mode (which applies, in particular, to conserved dynamics), we take Neumann modes for σ n [see Eq. (3.1c) [117]] and use standard Fourier-series relations [118] to evaluate Eq. (B6) with h = 0: This profile agrees with the one obtained within linear MFT [see Eq. (38) in [61]] and accordingly fulfills, in an averaged sense, (++) capillary BCs of critical adsorption, as well as L 0 dz φ(z) = 0. If, instead, h 1 = 0 and the OP field is only subject to a bulk-like field h = 0 at location R, Eq. (B6) renders In an analogous way, we obtain from Eq. (B5) the connected static correlation function [see Eqs. (3.14) and (3.15)]: where δφ ≡ φ− φ denotes the fluctuation part of φ and we used J * n = J −n (which follows from J(z) being real-valued) and [σ (p) −n (z). Note that C φ is independent of h and h 1 and thus applies irrespective of the presence of boundary fields or linearly coupled tracers. Explicit expressions for C φ are provided in Eq. (3.19). It is useful to note that the profile in Eq. (B7) can be written as which readily follows from Eqs. (B6) and (B10).

System with a quadratically coupled tracer
In the case of a quadratically coupled tracer, correlations can be determined analogously to Eq. (B10) by defining . We focus here on the variance φ(R) 2 of the field at the location of the tracer, which can be obtained from Eq. (5.1) as where we have used Eqs. (5.2) and (5.5) and omitted an (infinite) numerical prefactor which cancels out in Eq. (B12).

Appendix C: Adiabatic limit of the OP dynamics
In order to determine an approximation to the solution φ(z, t) [Eq. (3.10)] in the adiabatic limit (χ 1), we substitute in Eq. (3.10a) the integration variable s = t − uχ/(k 2 n k 2a n ) and obtain The dependence onχ can be made explicit by rescaling time accordingly, see the discussion around Eq. (2.8). Due to the exponential, the integrand gives substantial contributions only if u O(1). Accordingly, forχ 1 it is justified to Taylor expand the terms in the square bracket in Eq. (C1) up to first order in uχ/(k 2 k 2a ), rendering It is not feasible here to expand beyond O(u) or, correspondingly O(χ), since this would generate derivatives of the noise ξ n (t). We remark that, alternatively to deriving an equation of motion, the adiabatic approximation can also be applied on the level of the correlation function C φ in Eq. (3.15).
The steady-state solution P s (R) of Eq. (D1) is given by with a normalization constant Z. The drift typically takes the following generic form: where U(z) is a potential, D(z)/T represents a mobility and αD (z) is the "spurious" drift. Using Eq. (D4) in Eq. (D3) renders Accordingly, in order to recover the standard Boltzmann equilibrium distribution from the FPE in Eq. (D1), the drift in Eq. (D4) must involve a "spurious" contribution with α = 1 (isothermal convention) [83].

Determinant
The form of Eq. (G3) allows us to apply the matrix determinant lemma: For the purpose of regularization, we keep the mode number M finite and let M → ∞ only at the end of the calculation. Considering first BCs without a zero mode, it follows from the expression for the variance V φ (R) in Eq. (3.20) that Accordingly, Eq. (G4) evaluates to where ρ ≡ R/L and κ c = cLT φ /T R [see Eq. (4. 35)]. While the quantity N ≡ det(A) formally diverges for M → ∞, it is independent of R and c and thus canceled by the normalization of the distribution [see, e.g., Eq. (5.5)]. We obtain and N (p) = 2 2M N (N * ) 2 . For BCs with a zero mode, Eq.
which is independent of R.

Inverse
According to the Sherman-Morrison formula [89], the inverse of Γ in Eq. (G3) is given by with (uu † ) nm = u n u * m and where the last equation follows from the fact that A is a real symmetric matrix. Upon introducing the static correlation function C φ via Eq. (3.19), one obtains: In the specific case of Neumann BCs with a zero mode we also need the following expression: The calculation of the equilibrium distributions for d = 1 in Sec. V A is extended here to a film geometry in d > 1 dimensions. Where necessary, we specialize the calculation to d = 3. The film is assumed to have boundaries at z = 0, L and is macroscopically extended (transverse area A) in the other directions. For the purpose of regularization, we first consider a finite A and perform the thin film limit A → ∞ at the end of the calculation.
Inserting the standard eigenmode expansion of the OP, φ(r , z) = 1 √ A n,p σ n (z)e ip·r φ n (p), withΓ n,p,m,q (R z ) = 1 T φ (k 2 n + p 2 )δ n,m δ p,q + c AT R e ip·R σ n (R z )e −iq·R σ −m (R z ), wherein R z denotes the z-coordinate of the tracer position. In writing the last term inΓ, we used the fact that φ −n (−p) = φ * n (p) for a real-valued φ(r), such that one recovers the expected contribution c 2A m,n p,q e ip·R +iq·R × σ n (R z )σ m (R z )φ n (p)φ m (q) in the Hamiltonian. We now separately discuss the case of a linearly and a quadratically coupled tracer.

Linearly coupled tracer
We assume h = 0, h 1 = c = 0 and perform the Gaussian integration over the OP field [see Eqs. (2.12) and (5.2)], which yields the equilibrium distribution of R z : with where Ω d is the surface area of the d-dimensional unit sphere. In Eq. (H4), we have performed the thin film limit A → ∞ by applying the rule where λ and Λ are low-and high-wavenumber cutoffs introduced for the purpose of regularization. Note that the expression for Φ in Eq. (H4) essentially corresponds to the variance of the OP φ(R , R z ) in d dimensions, i.e., Φ(R z ) = h 2 In the case of periodic BCs, we accordingly obtain a spatially uniform distributionP s . For the other BCs, we get Λ λ dp p d−2 1 ∓ cos(2πnR z /L) (πn/L) 2 + p 2 , where the − (+) sign applies to Dirichlet (Neumann) BCs. In order to calculate Eq. (H8) in d = 3, we first determine the sum over n (see §1.445 in [118]) and subsequently integrate over p. The result can be written in the form Φ(R z ) =Φ(R z ) + Φ 0 , where Φ 0 is a R z -independent term that diverges ∼ λ d−3 (∼ log λ in d = 3) and ∼ Λ d−1 , while the R z -dependent partΦ(R z ) is independent of the cutoffs. Since all R z -independent parts are canceled by the normalization in Eq. (H4), one may omit them inΦ and accordingly obtain where ψ(z) = Γ (z)/Γ(z) is the digamma function [123] and is a dimensionless coupling [cf. Eq. (4. 3)]. The normalization factor Z in Eq. (H4) has to be computed numerically.
The functionΦ diverges at the boundaries:Φ (analogously for R z → L), renderingP s non-normalizable for Neumann BCs. This divergence ofΦ is cut off if Λ is finite. In a physical system, van der Waals interactions regularize the singular Casimir potential near the boundary [6]. Instead ofP s , we thus show in Fig. 17 the quantityΦ, which essentially corresponds to the negative Casimir potential defined in Eq. (6.4), U(R z ) = −T Φ(R z ) (setting T = T R = T φ ).

Quadratic coupling
For c = 0 and h = h 1 = 0, a Gaussian integration over the field modes using Eq. (H2) yields [cf. Eq. (5.5)] Upon expressingΓ in terms of (u) np = c/(AT R )e ip·R σ n (R z ) np and A np,mq = δ n,m δ p,q (k 2 n + p 2 )/T φ , the determinant can be calculated using the formalism in Appendix G. In the thin film limit, this results in [see Eq. (G4)] which has essentially been calculated after Eq. (H4). Accordingly, for periodic BCs, it is independent of R z , whereas for Neumann or Dirichlet BCs, the analysis around Eq. (H8) implies that it is dominated by an R z -independent As discussed above [see Eq. (H14)], in the continuum limit Λ → ∞, the contribution of the determinant is dominated by a divergent term and is thus canceled by the normalization factor Z. The exponent in Eq. (H16) coincides (apart from the prefactor A) with its one-dimensional counterpart in Eq. (5.13) and, taking Neumann modes for σ n and using Appendix G, we obtain (with ρ ≡ R z /L) (±c±) (H17) The (dimensionless) effective couplings H 1 ≡ h 1 L d /T φ and κ c ≡ cL 2−d T φ /T R generalize the ones in Eqs. (4.35) and (4.41) to d dimensions. The expression in Eq. (H17) differs from Eq. (5.14) by the absence of the R z -dependent prefactor stemming from the determinant. In the limit κ c → ∞, the probability distribution is given bȳ which is independent of κ c . The normalization factors Z in Eqs. (H17) and (H18) have to be determined numerically.