Table of contents

We prove the Jarzynski relation for general stochastic processes including non-Markovian systems with memory. The only requirement for our proof is the existence of a stationary state, therefore excluding non-ergodic systems. We then show how the concepts of stochastic thermodynamics can be used to prove further exact non-equilibrium relations like the Crooks relation and the fluctuation theorem on entropy production for non-Markovian dynamics.

Metapopulation models describing cities with different populations coupled by the travel of individuals are of great importance in the understanding of disease spread on a large scale. An important example is the Rvachev–Longini model which is widely used in computational epidemiology. Few analytical results are, however, available and, in particular, little is known about paths followed by epidemics and disease arrival times. We study the arrival time of a disease in a city as a function of the starting seed of the epidemics. We propose an analytical ansatz, test it in the case of a spread on the worldwide air-transportation network, and show that it predicts accurately the arrival order of a disease in worldwide cities.

We study the dynamics of a single-chain polymer confined to a two-dimensional cell. We introduce a kinetically constrained lattice gas model that preserves the connectivity of the chain, and we use this kinetically constrained model to study the dynamics of the polymer at varying densities through Monte Carlo simulations. Even at densities close to the fully packed configuration, we find that the monomers comprising the chain manage to diffuse around the box with a root mean square displacement of the order of the box dimensions over timescales for which the overall geometry of the polymer is, nevertheless, largely preserved. To capture this shape persistence, we define the local tangent field and study the two-time tangent–tangent correlation function, which exhibits a glass-like behavior. In both closed and open chains, we observe reptational motion and reshaping through local fingering events which entail global monomer displacement.

We demonstrate that the transfer matrix of the inhomogeneous N-state chiral Potts model with two vertical superintegrable rapidities serves as the Q operator of the XXZ chain model for a cyclic representation of with Nth root-of-unity and representation parameter for odd N. The symmetry problem of the XXZ chain with a general cyclic representation is mapped onto the problem of studying the Q operator of some special one-parameter family of generalized τ⁽²⁾ models. In particular, the spin-(N−1)/2 XXZ chain model with and the homogeneous N-state chiral Potts model at a specific superintegrable point are unified as one physical theory. By Baxter's method, developed for producing a Q₇₂ operator of the root-of-unity eight-vertex model, we construct the Q_R,Q_L and Q operators of a superintegrable τ⁽²⁾ model, then identify them with transfer matrices of the N-state chiral Potts model for a positive integer N. We thus obtain a new method of producing the superintegrable N-state chiral Potts transfer matrix from the τ⁽²⁾ model by constructing its Q operator.

We analyze a collaboration network based on the Marvel Universe comic books. First, we consider the system as a binary network, where two characters are connected if they appear in the same publication. The analysis of degree correlations reveals that, in contrast to most real social networks, the Marvel Universe presents a disassortative mixing on the degree. Then, we use a weight measure to study the system as a weighted network. This allows us to find and characterize well defined communities. Through the analysis of the community structure and the clustering as a function of the degree we show that the network presents a hierarchical structure. Finally, we comment on possible mechanisms responsible for the particular motifs observed.

This paper presents an exact probabilistic description of the work done by an external agent on a two-level system. We first develop a general scheme which is suitable for the treatment of functionals of the time-inhomogeneous Markov processes. Subsequently, we apply the procedure to the analysis of the isothermal-work probability density and we obtain its exact analytical forms in two specific settings. In both models, the state energies change with a constant velocity. On the other hand, the two models differ in their interstate transition rates. The explicit forms of the probability density allow a detailed discussion of the mean work. Moreover, we discuss the weight of the trajectories which display a smaller value of work than the corresponding equilibrium work. The results are controlled by a single dimensionless parameter which expresses the ratio of two underlying timescales: the velocity of the energy changes and the relaxation time in the case of frozen energies. If this parameter is large, the process is a strongly irreversible one and the work probability density differs substantially from a Gaussian curve.

We study experimentally the thermal fluctuations of energy input and dissipation in a harmonic oscillator driven out of equilibrium, and search for fluctuation relations. Both the transient evolution from the equilibrium state, and non-equilibrium steady states are analyzed. Fluctuation relations are obtained experimentally for both the work and the heat, for the stationary and transient evolutions. A stationary state fluctuation theorem is verified for various time dependences of the imposed external torque. The transient fluctuation theorem is satisfied for the work given to the system but not for the heat dissipated by the system in the case of linear forcing. Experimental observations on the statistical and dynamical properties of the position fluctuations of the torsion pendulum allow us to derive analytical expressions for the probability density functions of the work and the heat. We obtain for the first time an analytic expression for the probability density function of the heat. The agreement between experiments and our predictions is excellent.

We develop a variational scheme in a field theoretic approach to a stochastic process. While various stochastic processes can be expressed using master equations, in general it is difficult to solve the master equations exactly, and it is also hard to solve the master equations numerically because of the curse of dimensionality. The field theoretic approach has been used in order to study such complicated master equations, and the variational scheme achieves tremendous reduction in the dimensionality of master equations. For the variational method, only the Poisson ansatz has been used, in which one restricts the variational function to a Poisson distribution. Hence, one has dealt with only restricted fluctuation effects. We develop the variational method further, which enables us to treat an arbitrary variational function. It is shown that the variational scheme developed gives a quantitatively good approximation for master equations which describe a stochastic gene regulatory network.

We investigate a lattice model of polymers where the nearest neighbour monomer–monomer interaction strengths differ according to whether the local configurations have so-called 'hydrogen-like' formations or not. If the interaction strengths are all the same then the classical θ-point collapse transition occurs on lowering the temperature, and the polymer enters the isotropic liquid drop phase known as the collapsed globule. On the other hand, strongly favouring the hydrogen-like interactions gives rise to an anisotropic folded (solid-like) phase on lowering the temperature. We use Monte Carlo simulations up to a length of 256 to map out the phase diagram in the plane of parameters and determine the order of the associated phase transitions. We discuss the connections to semi-flexible polymers and other polymer models. Importantly, we demonstrate that for a range of energy parameters, two phase transitions occur on lowering the temperature, the second being a transition from the globule state to the crystal state. We argue from our data that this globule-to-crystal transition is continuous in two dimensions in accord with field-theory arguments concerning Hamiltonian walks, but is first order in three dimensions.

We present an experimental study of the kinetic and structural aspects of heteroaggregation in binary dipolar monolayers. Particles of the two components of the system have oppositely oriented dipole moments constrained to be perpendicular to the plane of motion. We show experimentally that growing clusters have chain-like morphology with a crossover to branched structures, whose fractal dimension is equal to that of self-avoiding random walks and reaction-limited cluster–cluster aggregation, respectively. The average cluster size and number of clusters have a power-law behavior as a function of time, in which the dynamic exponents depend on the concentration. Our experiments revealed a clear discrimination between clusters of an even and odd number of particles, i.e. the concentration of even-sized clusters has a much faster decay with time than that of the odd ones; furthermore, at a given time the concentration of clusters shows even–odd oscillations.

We study the metal–insulator transition of the one-dimensional diagonal Anderson ternary model with long range correlated disorder. The starting point of the model corresponds to a ternary alloy (i.e. with three possible on-site energies), and shows a metal–insulator transition when the random distribution of site energies is assumed to have a power spectrum . In this paper, we define a purity parameter for the ternary alloy which adjusts the occupancy probability of site potentials, and for any given α we calculate the critical purity parameter for which extended states are obtained. In this way, we show that the ternary alloy requires weaker correlations than the binary alloy to present a phase transition from localized to extended states. A phase diagram which separates the extended regime from the localized one for the ternary alloy is presented, obtained as the critical purity parameter in terms of the corresponding correlation exponent.

We continue our investigation of the two-dimensional Abelian sandpile model in terms of a logarithmic conformal field theory with central charge c = −2, by introducing two new boundary conditions. These have two unusual features: they carry an intrinsic orientation, and, more strangely, they cannot be imposed uniformly on a whole boundary (like the edge of a cylinder). They lead to seven new boundary condition changing fields, some of them being in highest weight representations (weights −1/8, 0 and 3/8), some others belonging to indecomposable representations with rank 2 Jordan cells (lowest weights 0 and 1). Their fusion algebra appears to be in full agreement with the fusion rules conjectured by Gaberdiel and Kausch.

We consider the application of fluctuation relations to the dynamics of coarse-grained systems, as might arise in a hypothetical experiment in which a system is monitored with a low resolution measuring apparatus. We analyze a stochastic, Markovian jump process with a specific structure that lends itself naturally to coarse-graining. A perturbative analysis yields a reduced stochastic jump process that approximates the coarse-grained dynamics of the original system. This leads to a non-trivial fluctuation relation that is approximately satisfied by the coarse-grained dynamics. We illustrate our results by computing the large deviations of a particular stochastic jump process. Our results highlight the possibility that observed deviations from fluctuation relations might be due to the presence of unobserved degrees of freedom.

We study numerically a two-dimensional monodisperse model of interacting classical particles predicted to exhibit a static liquid–glass transition. Using a dynamical Monte Carlo method we show that the model does not freeze into a glassy phase at low temperatures. Instead, depending on the choice of the hard-core radius for the particles, the system either collapses trivially or a polycrystalline hexagonal structure emerges.

Recently, it has been shown that, when the dimension of a graph turns out to be infinite-dimensional in a broad sense, the upper critical surface and the corresponding critical behavior of an arbitrary Ising spin glass model defined over such a graph can be exactly mapped on the critical surface and behavior of a non-random Ising model. A graph can be infinite-dimensional in a strict sense, like the fully connected graph, or in a broad sense, as happens on a Bethe lattice and in many random graphs. In this paper, we firstly introduce our definition of dimensionality which is compared to the standard definition and readily applied to test the infinite dimensionality of a large class of graphs which, remarkably enough, includes even graphs where the tree-like approximation (or, in other words, the Bethe–Peierls approach), in general, may be wrong. Then, we derive a detailed proof of the mapping for all the graphs satisfying this condition. As a by-product, the mapping provides immediately a very general Nishimori law.

We consider the open spin-sXXZ quantum spin chain with N sites and general integrable boundary terms for generic values of the bulk anisotropy parameter, and for values of the boundary parameters which satisfy a certain constraint. We derive two sets of Bethe ansatz equations, and find numerical evidence that together they give the complete set of (2s+1)^N eigenvalues of the transfer matrix. For the case of s = 1, we explicitly determine the Hamiltonian, and find an expression for its eigenvalues in terms of Bethe roots.

We study the non-equilibrium relaxation of an elastic line described by the Edwards–Wilkinson equation. Although this model is the simplest representation of interface dynamics, we highlight that many (though not all) important aspects of the non-equilibrium relaxation of elastic manifolds are already present in such quadratic and clean systems. We analyze in detail the ageing behavior of several two-times averaged and fluctuating observables taking into account finite size effects and the crossover to the stationary and equilibrium regimes. We start by investigating the structure factor and extracting from its decay a growing correlation length. We present the full two-times and size dependence of the interface roughness and we generalize the Family–Vicsek scaling form to non-equilibrium situations. We compute the incoherent scattering function and we compare it to the one measured for other glassy systems. We analyze the response functions, the violation of the fluctuation-dissipation theorem in the ageing regime, and its crossover to the equilibrium relation in the stationary regime. Finally, we study the out-of-equilibrium fluctuations of the previously studied two-times functions and we characterize the scaling properties of their probability distribution functions. Our results allow us to obtain new insights into other glassy problems such as the ageing behavior in colloidal glasses and vortex glasses.

We calculate the bulk contribution for the doubly degenerate largest eigenvalue of the transfer matrix of the eight-vertex model with an odd number of lattice sites N in the disordered regime using the generic equation for roots proposed by Fabricius and McCoy. We show that, as expected, in the thermodynamic limit the result coincides with the one in the N even case.

The transfer matrix of the XXZ open spin-½ chain with general integrable boundary conditions and generic anisotropy parameter (q is not a root of unity and |q| = 1) is diagonalized using the representation theory of the q-Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the complete spectrum is expressed in terms of the roots of a characteristic polynomial of degree d = 2^N. The complete family of eigenstates are derived in terms of rational functions defined on a discrete support which satisfy a system of coupled recurrence relations. In the special case of linear relations between left and right boundary parameters for which Bethe-type solutions are known to exist, our analysis provides an alternative derivation of the results of Nepomechie et al and Cao et al. In the latter case the complete family of eigenvalues and eigenstates splits into two sets, each associated with a characteristic polynomial of degree d < 2^N. Numerical checks performed for small values of N support the analysis.

A brief survey of the theoretical, numerical and experimental studies of the random field Ising model (RFIM) during the last three decades is given. The nature of the phase transition in the three-dimensional RFIM with Gaussian random fields is discussed. Using simple scaling arguments it is shown that if the strength of the random fields is not too small (bigger than a certain threshold value), the finite temperature phase transition in this system is equivalent to the low temperature order–disorder transition which takes place with variations of the strength of the random fields. A detailed study of the zero-temperature phase transition in terms of simple probabilistic arguments and a modified mean field approach (which take into account nearest neighbor spin–spin correlations) is given. It is shown that if all thermally activated processes are suppressed, the ferromagnetic order parameter m(h) as a function of the strength h of the random fields becomes history dependent. In particular, the behavior of the magnetization curves m(h) for increasing and decreasing h reveals a hysteresis loop.

In this paper, we investigate the escape for the mean first passage time (MFPT) over the fluctuating potential barrier for a system only driven by a three-state Markovian noise. It is shown that, in some circumstances, the three-state Markovian noise can make the particles escape over the fluctuating potential barrier, but in other circumstances it cannot. We find four resonant activations for the MFPT over the fluctuating potential barrier, which are, respectively, the functions of the flipping rate of the fluctuating potential barrier and the three transition rates of the three-state Markovian noise considered by us.

We study a model eco-system by means of dynamical techniques from disordered systems theory. The model describes a set of species subject to competitive interactions through a background of resources, which they feed upon. Additionally direct competitive or co-operative interaction between species may occur through a random coupling matrix. We compute the order parameters of the system in a fixed point regime and identify the onset of instability and compute the phase diagram. We focus on the effects of variability of resources, direct interaction between species, co-operation pressure and dilution on the stability and the diversity of the eco-system. It is shown that resources can be exploited optimally only in the absence of co-operation pressure or direct interaction between species.

We present an explicit conjecture for the chiral fusion algebra of critical percolation considering Virasoro representations with no enlarged or extended symmetry algebra. The representations that we take to generate fusion are countably infinite in number. The ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of these representations decomposes into a finite direct sum of these representations. The fusion rules are commutative, associative and exhibit an structure. They involve representations which we call Kac representations of which some are reducible yet indecomposable representations of rank 1. In particular, the identity of the fusion algebra is a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the recent results of Eberle–Flohr and Read–Saleur. Notably, in agreement with Eberle–Flohr, we find the appearance of indecomposable representations of rank 3. Our fusion rules are supported by extensive numerical studies of an integrable lattice model of critical percolation. Details of our lattice findings and numerical results will be presented elsewhere.

Irreversible opinion spreading phenomena are studied on small-world networks generated from 2D regular lattices by means of the magnetic Eden model, a non-equilibrium kinetic model for the growth of binary mixtures in contact with a thermal bath. In this model, the opinion or decision of an individual is affected by those of their acquaintances, but opinion changes (analogous to spin flips in an Ising-like model) are not allowed. In particular, we focus on aspects inherent to the underlying 2D nature of the substrate, such as domain growth and cluster size distributions. Larger shortcut fractions are observed to favor long-range ordering connections between distant clusters across the network, while the temperature is shown to drive the system across an order–disorder transition, in agreement with previous investigations on related equilibrium spin systems. Furthermore, the extrapolated phase diagram, as well as the correlation length critical exponent, are determined by means of standard finite-size scaling procedures.