Statistical Physics special collection

In elementary textbooks, the microscopic justification of Ohm's local law in a solid medium starts with Drude's classical model of electron transport and next discusses the quantum-dynamical and statistical amendments. In this paper, emphasis is laid instead upon the thermodynamical background motivated by the Joule–Lenz heating effect accompanying conduction and the fact that the conduction electrons are thermalized at the lattice temperature. Both metals and n-type semiconductors are considered; but conduction under a magnetic field is not. Proficiency in second-year thermodynamics and vector analysis is required from an undergraduate university student in physics so that the content of the paper can be taught to third-year students. The necessary elements of quantum mechanics are posited in this paper without detailed justification. We start with the equilibrium-thermodynamic notion of the chemical potential of the electron gas, the value of which distinguishes metals from semiconductors. Then we turn to the usage of the electrochemical potential in the description of near-equilibrium electron transport. The response of charge carriers to the electrochemical gradient involves the mobility, which is the reciprocal of the coefficient of the effective friction force opposing the carrier drift. Drude's calculation of mobility is restated with the dynamical requirements of quantum physics. Where the carrier density is inhomogeneous, there appears diffusion, the coefficient of which is thermodynamically related to the mobility. Next, it is remarked that the release of heat was ignored in Drude's original model. In this paper, the flow of Joule heat is handled thermodynamically within an energy balance where the voltage generator, the conduction electrons and the host lattice are involved in an explicit way. The notion of dissipation is introduced as the rate of entropy creation in a steady state. The body of the paper is restricted to the case of one homogeneous temperature. The generalisation of the thermodynamical framework to an inhomogeneous temperature field is sketched in an appendix. A fluid-mechanical picture of electronic conduction is obtained as a by-product of that framework.

From cement cohesion to DNA condensation, a proper statistical physics treatment of systems with long-range forces is important for a number of applications in physics, chemistry, and biology. We compute here the effective force between fixed charged macromolecules, screened by oppositely charged mobile ions (counterions). We treat the problem in a one-dimensional configuration that allows for interesting discussion and derivation of exact results, remaining at a level of mathematical difficulty compatible with an undergraduate course. Emphasis is put on the counterintuitive but fundamental phenomenon of like-charge attraction, which our treatment brings for the first time to the level of undergraduate teaching. The parity of the number of counterions is shown to play a prominent role, which sheds light on the binding mechanism at work when like-charge macromolecules do attract.

Symmetries regarding system-surroundings interchange are used to propose the attribution of different meanings to the terms 'adiabatic' and 'thermally insulated' and address the resulting implications. It is also shown that entropy generation can be interpreted as the ratio of lost work by the temperature at which such loss occurs, and that it occurs always in the system.

In this paper the effect of the presence of temperature dependent transient thermodynamic processes such as the dissociation, condensation and excitation of vibrational degrees of freedom on the peculiarities of the behaviour of an imperfect gas was considered on a quantitative level. We also discussed the influence of the nonequilibrium thermodynamic effects on the behaviour of real gas and estimated the order of their magnitudes.

The statistical mechanics of small clusters (n ∼ 10–50 elements) of harmonic oscillators and two-level systems is studied exactly, following the microcanonical, canonical and grand canonical formalisms. For clusters with several hundred particles, the results from the three formalisms coincide with those found in the thermodynamic limit. However, for clusters formed by a few tens of elements, the three ensembles yield different results. For a cluster with a few tens of harmonic oscillators, when the heat capacity per oscillator is evaluated within the canonical formalism, it reaches a limit value equal to k_B, as in the thermodynamic case, while within the microcanonical formalism the limit value is k_B(1–1/n). This difference could be measured experimentally. For a cluster with a few tens of two-level systems, the heat capacity evaluated within the canonical and microcanonical ensembles also presents differences that could be detected experimentally. Both the microcanonical and grand canonical formalism show that the entropy is non-additive for systems this small, while the canonical ensemble reaches the opposite conclusion. These results suggest that the microcanonical ensemble is the most appropriate for dealing with systems with tens of particles.

The entropy of the Ising model in the mean field approximation is derived by the Hamilton–Jacobi formalism. We consider a grand canonical ensemble with respect to the temperature and the external magnetic field. A cusp arises at the critical point, which shows a simple and new geometrical aspect of this model. In an educational sense, this curve with a cusp helps students acquire a more intuitive view of statistical phase transitions.

In 1867, just two years after laying the foundations of electromagnetism, J. Clerk Maxwell presented a fundamental paper on kinetic gas theory, in which he described the evolution of the gas in terms of certain 'moments' of its velocity distribution function. This inspired Ludwig Boltzmann to formulate his famous kinetic equation, from which followed the H-theorem and the connection with entropy. On the occasion of the 150th anniversary of publication of Maxwell's paper, we review the Maxwell–Boltzmann formalism and discuss how its generality and adaptability enable it to play a key role in describing the behaviour of a variety of systems of current interest, in both gaseous and condensed matter, and in modern-day physics and technologies which Maxwell and Boltzmann could not possibly have foreseen. In particular, we illustrate the relevance and applicability of Maxwell's formalism to the dynamic field of plasma-wakefield acceleration.

The motion of a particle described by the Langevin equation with constant diffusion coefficient, time dependent linear force ($\omega (1+\alpha \cos ({\omega }_{1}t))x$) and periodic load force (${A}_{0}\cos ({\rm{\Omega }}t)$) is investigated. Analytical solutions for the probability density function (PDF) and n-moment are obtained and analysed. For ${\omega }_{1}\gg \alpha \omega$ the influence of the periodic term $\alpha \cos ({\omega }_{1}t)$ is negligible to the PDF and n-moment for any time; this result shows that the statistical averages such as n-moments and the PDF have no access to some information of the system. For small and intermediate values of ${\omega }_{1}$ the influence of the periodic term $\alpha \cos ({\omega }_{1}t)$ to the system is also analysed; in particular the system may present multiresonance. The solutions are obtained in a direct and pedagogical manner readily understandable by graduate students.

Osmosis drives the development of a pressure difference of many atmospheres between a dilute solution and pure solvent with which it is in contact through a semi-permeable membrane. The educational importance of this paper is that it presents a novel treatment in terms of fluid mechanics that is quantitative and exact. It is also simple and intuitive, showing vividly how osmotic pressures are generated and maintained in equilibrium, driven by differential solvent pressures. The present rigorous analysis using the virial theorem seems unknown and can be easily understood—and taught—at various different levels. It should be valuable to undergraduates, graduate students and indeed to the general physicist.

Superfluid helium, a state of matter existing at low temperatures, shows many remarkable properties. One example is the so called fountain effect, where a heater can produce a jet of helium. This converts heat into mechanical motion; a machine with no moving parts, but working only below 2 K. Allen and Jones first demonstrated the effect in 1938, but their work was basically qualitative. We now present data of a quantitative version of the experiment. We have measured the heat supplied, the temperature and the height of the jet produced. We also develop equations, based on the two-fluid model of superfluid helium, that give a satisfactory fit to the data. The experiment has been performed by advanced undergraduate students in our home institution, and illustrates in a vivid way some of the striking properties of the superfluid state.

In this article we present an educational approach to thermal equilibrium which was tested on a group of 13 undergraduate students at the University of Trento. The approach is based on a stochastic toy model, in which bodies in thermal contact are represented by rows of squares on a cardboard table, which exchange coins placed on the squares based on the roll of two dice. The discussion of several physical principles, such as the exponential approach to equilibrium, the determination of the equilibrium temperature, and the interpretation of the equilibrium state as the most probable macrostate, proceeds through a continual comparison between the outcomes obtained with the toy model and the results of a real experiment on the thermal contact of two masses of water at different temperatures. At the end of the sequence, a re-analysis of the experimental results in view of both the Boltzmann and Clausius definitions of entropy reveals some limits of the toy model, but also allows for a critical discussion of the concepts of temperature and entropy. In order to provide the reader with a feeling of how the sequence was received by students, and how it helped them understand the topics introduced, we discuss some excerpts from their answers to a conceptual item given at the end of the sequence.

Here we discuss blackbody radiation within the context of classical theory. We note that nonrelativistic classical mechanics and relativistic classical electrodynamics have contrasting scaling symmetries which influence the scattering of radiation. Also, nonrelativistic mechanical systems can be accurately combined with relativistic electromagnetic radiation only provided the nonrelativistic mechanical systems are the low-velocity limits of fully relativistic systems. Application of the no-interaction theorem for relativistic systems limits the scattering mechanical systems for thermal radiation to relativistic classical electrodynamic systems, which involve the Coulomb potential. Whereas the naive use of nonrelativistic scatterers or nonrelativistic classical statistical mechanics leads to the Rayleigh–Jeans spectrum, the use of fully relativistic scatterers leads to the Planck spectrum for blackbody radiation within classical physics.

To illustrate a simple mean-field-like approach for examining quantum phase transitions we consider the $J\mbox{--}{J}^{\prime }$ quantum Heisenberg antiferromagnet on a square lattice. The exchange couplings J and ${J}^{\prime }$ are competing with each other. The ratio ${J}^{\prime }/J$ is the control parameter and its change drives the transition. We adopt a variational ansatz, calculate the ground-state energy as well as the order parameter and describe the quantum phase transition inherent in the model. This description corresponds completely to the standard Landau theory of phase transitions. We also discuss how to generalize such an approach for more complicated quantum spin models.

Water, which is the habitat for a variety of living creatures, has a maximum density at 4.0 °C. This crucial property is considered to play a very important role in the biology of a lake and also has a close relationship with the areas of environmentology and geoscience. It would be desirable for students to confirm this important property of water themselves by carrying out simple experiments. However, it is not easy to detect the maximum density at 4.0 °C because the temperature dependence of the water density is very small close to its freezing point. For example, the density of water is 0.999 975 g cm⁻³ at 4.0 °C and 0.999 850 g cm⁻³ at 0.1 °C. The aim in this manuscript is to demonstrate a simple experiment to detect 4.0 °C as the temperature of maximum density, in which the time dependence of the water temperature is measured at several different depths by chilling the water surface. This is a simple experiment that can also be performed by high school students. We also present a mathematical model that can explain the results of this experiment.

With special attention to the role of information theory in physical sciences we present analytical results for the coordinate- and momentum-space Fisher information of some important one-dimensional quantum systems which differ in spacing of their energy levels. The studies envisaged allow us to relate the coordinate-space information (${I}_{\rho }$) with the familiar energy levels of the quantum system. The corresponding momentum-space information (${I}_{\gamma }$) does not obey such a simple relationship with the energy spectrum. Our results for the product (${I}_{\rho }{I}_{\gamma }$) depend quadratically on the principal quantum number n and satisfy an appropriate uncertainty relation derived by Dehesa et al (2007 J. Phys. A: Math. Theor. 40 1845)

This paper shows that when students are introduced to the derivation of one of the most important physical formulas—the canonical distribution—they are exposed to assumptions which may be confusing because they contradict physical reality. The paper provides an alternative derivation of the canonical distribution. Our derivation takes into account internal physical processes in macrosystems leading to the canonical distribution and does not require any physically unjustified assumptions. The article is intended for teachers and students of statistical physics, and for general physicists.

Complex systems are characterised by specific time-dependent interactions among their many constituents. As a consequence they often manifest rich, non-trivial and unexpected behaviour. Examples arise both in the physical and non-physical worlds. The study of complex systems forms a new interdisciplinary research area that cuts across physics, biology, ecology, economics, sociology, and the humanities. In this paper we review the essence of complex systems from a physicists' point of view, and try to clarify what makes them conceptually different from systems that are traditionally studied in physics. Our goal is to demonstrate how the dynamics of such systems may be conceptualised in quantitative and predictive terms by extending notions from statistical physics and how they can often be captured in a framework of co-evolving multiplex network structures. We mention three areas of complex-systems science that are currently studied extensively, the science of cities, dynamics of societies, and the representation of texts as evolutionary objects. We discuss why these areas form complex systems in the above sense. We argue that there exists plenty of new ground for physicists to explore and that methodical and conceptual progress is needed most.

The Ising model is a well known statistical model which can be solved exactly by various methods. The most familiar one is the transfer matrix method. Sometimes it can be difficult to approach the open boundary case rather than periodic boundary ones in higher dimensions. But physically it is more intuitive to study the open boundary case, as it gives a closer view of the real system. We have introduced a new method called the pairing method to determine the exact partition function for the simplest case, a 1D Ising lattice. This method simplifies the problem's complexities and reduces it to a pure combinatorial problem. The study also reveals that it is possible to apply this pairing method in the case of a 2D square lattice. The obtained results agree perfectly with the values in the literature and this new approach provides an algorithmic insight to deal with such problems.

In undergraduate statistical mechanics courses the Ising model always plays an important role because it is the simplest non-trivial model used to describe magnetic systems. The one-dimensional model is easily solved analytically, while the two-dimensional one can be solved exactly by the Onsager solution. For this reason, numerical simulations are usually used to solve the two-dimensional model. Keeping in mind that the two-dimensional model is the platform for studying phase transitions, it is usually an exercise in computational undergraduate courses because its numerical solution is relatively simple to implement and its critical exponents are perfectly known. The purpose of this article is to present a detailed numerical study of the second-order phase transition in the two-dimensional Ising model at an undergraduate level, allowing readers not only to compare the mean-field solution, the exact solution and the numerical one through a complete study of the order parameter, the correlation function and finite-size scaling, but to present the techniques, along with hints and tips, for solving it themselves. We present the elementary theory of phase transitions and explain how to implement Markov chain Monte Carlo simulations and perform them for different lattice sizes with periodic boundary conditions. Energy, magnetization, specific heat, magnetic susceptibility and the correlation function are calculated and the critical exponents determined by finite-size scaling techniques. The importance of the correlation length as the relevant parameter in phase transitions is emphasized.

An approximate theory is presented that describes the propagation of the ice–water front that develops in droplets of water that are deposited on a planar surface at a temperature below the melting point of ice. This theory is compared with experimental observation of the time evolution of this front. These experiments were performed by freezing water droplets directly on a block of dry ice, and to examine the effects of the thermal conductivity of a substrate during the freezing process. Such droplets were also deposited on a glass plate and on a copper plate placed on dry ice. The temperature at the base of these droplets, and the dependence of the freezing time on their size were also obtained experimentally, and compared with our analytic theory. These experiment can be readily performed by physics undergraduate students, and reveal that the usual assumption of constant temperature at the base of the droplets cannot be implemented in practice.