The Nonequilibrium thermodynamics of small systems

The interactions of tiny objects with their environments are dominated by thermal fluctuations. Guided by theory and assisted by new micromanipulation tools, scientists have begun to study such interactions in detail.

Diffusion in stationary flow from mesoscopic nonequilibrium thermodynamics

We analyze the diffusion of a Brownian particle in a fluid under stationary flow. By using the scheme of nonequilibrium thermodynamics in phase space, we obtain the Fokker-Planck equation that is compared with others derived from the kinetic theory and projector operator techniques. This equation exhibits violation of the fluctuation-dissipation theorem. By implementing the hydrodynamic regime described by the first moments of the nonequilibrium distribution, we find relaxation equations for the diffusion current and pressure tensor, allowing us to arrive at a complete description of the system in the inertial and diffusion regimes. The simplicity and generality of the method we propose makes it applicable to more complex situations, often encountered in problems of soft-condensed matter, in which not only one but more degrees of freedom are coupled to a nonequilibrium bath.

Critical review of coupled flux formulations for clay membranes based on nonequilibrium thermodynamics

Extensive research conducted over the past several decades has indicated that semipermeable membrane behavior (i.e., the ability of a porous medium to restrict the passage of solutes) may have a significant influence on solute migration through a wide variety of clay-rich soils, including both natural clay formations (aquitards, aquicludes) and engineered clay barriers (e.g., landfill liners and vertical cutoff walls). Restricted solute migration through clay membranes generally has been described using coupled flux formulations based on nonequilibrium (irreversible) thermodynamics. However, these formulations have differed depending on the assumptions inherent in the theoretical development, resulting in some confusion regarding the applicability of the formulations. Accordingly, a critical review of coupled flux formulations for liquid, current, and solutes through a semipermeable clay membrane under isothermal conditions is undertaken with the goals of explicitly resolving differences among the formulations and illustrating the significance of the differences from theoretical and practical perspectives. Formulations based on single-solute systems (i.e., uncharged solute), single-salt systems, and general systems containing multiple cations or anions are presented. Also, expressions relating the phenomenological coefficients in the coupled flux equations to relevant soil properties (e.g., hydraulic conductivity and effective diffusion coefficient) are summarized for each system. A major difference in the formulations is shown to exist depending on whether counter diffusion or salt diffusion is assumed. This difference between counter and salt diffusion is shown to affect the interpretation of values for the effective diffusion coefficient in a clay membrane based on previously published experimental data. Solute transport theories based on both counter and salt diffusion then are used to re-evaluate previously published column test data for the same clay membrane. The results indicate that, despite the theoretical inconsistency between the counter-diffusion assumption and the salt-diffusion conditions of the experiments, the predictive ability of solute transport theory based on the assumption of counter diffusion is not significantly different from that based on the assumption of salt diffusion, provided that the input parameters used in each theory are derived under the same assumption inherent in the theory. Nonetheless, salt-diffusion theory is fundamentally correct and, therefore, is more appropriate for problems involving salt diffusion in clay membranes. Finally, the fact that solute diffusion cannot occur in an ideal or perfect membrane is not explicitly captured in any of the theoretical expressions for total solute flux in clay membranes, but rather is generally accounted for via inclusion of an effective porosity, n(e), or a restrictive tortuosity factor, tau(r), in the formulation of Fick's first law for diffusion. Both n(e) and tau(r) have been correlated as a linear function of membrane efficiency. This linear correlation is supported theoretically by pore-scale modeling of solid-liquid interactions, but experimental support is limited. Additional data are needed to bolster the validity of the linear correlation for clay membranes. Copyright 2012 Elsevier B.V. All rights reserved.

Critical Review of Coupled Flux Formulations for Clay Membranes Based on Nonequilibrium Thermodynamics

Extensive research conducted over the past several decades has indicated that semipermeable membrane behavior (i.e., the ability of a porous medium to restrict the passage of solutes) may have a significant influence on solute migration through a wide variety of clay-rich soils, including both natural clay formations (aquitards, aquicludes) and engineered clay barriers (e.g., landfill liners and vertical cutoff walls). Restricted solute migration through clay membranes generally has been described using coupled flux formulations based on nonequilibrium (irreversible) thermodynamics. However, these formulations have differed depending on the assumptions inherent in the theoretical development, resulting in some confusion regarding the applicability of the formulations. Accordingly, a critical review of coupled flux formulations for liquid, current, and solutes through a semipermeable clay membrane under isothermal conditions is undertaken with the goals of explicitly resolving differences among the formulations and illustrating the significance of the differences from theoretical and practical perspectives. Formulations based on single-solute systems (i.e., uncharged solute), single-salt systems, and general systems containing multiple cations or anions are presented. Also, expressions relating the phenomenological coefficients in the coupled flux equations to relevant soil properties (e.g., hydraulic conductivity and effective diffusion coefficient) are summarized for each system. A major difference in the formulations is shown to exist depending on whether counter diffusion or salt diffusion is assumed. This difference between counter and salt diffusion is shown to affect the interpretation of values for the effective diffusion coefficient in a clay membrane based on previously published experimental data. Solute transport theories based on both counter and salt diffusion then are used to re-evaluate previously published column test data for the same clay membrane. The results indicate that, despite the theoretical inconsistency between the counter-diffusion assumption and the salt-diffusion conditions of the experiments, the predictive ability of solute transport theory based on the assumption of counter diffusion is not significantly different from that based on the assumption of salt diffusion, provided that the input parameters used in each theory are derived under the same assumption inherent in the theory. Nonetheless, salt-diffusion theory is fundamentally correct and, therefore, is more appropriate for problems involving salt diffusion in clay membranes. Finally, the fact that solute diffusion cannot occur in an ideal or perfect membrane is not explicitly captured in any of the theoretical expressions for total solute flux in clay membranes, but rather is generally accounted for via inclusion of an effective porosity, ne, or a restrictive tortuosity factor, tr, in the formulation of Fick's first law for diffusion. Both ne and tr have been correlated as a linear function of membrane efficiency. This linear correlation is supported theoretically by pore-scale modeling of solid-liquid interactions, but experimental support is limited. Additional data are needed to bolster the validity of the linear correlation for clay membranes.

Topics in equilibrium and nonequilibrium thermodynamics: computing crystalline free energies and engineering Maxwell’s demon.

This dissertation covers two separate topics in statistical physics. The first part of the dissertation focuses on computational methods of obtaining the free energies (or partition functions) of crystalline solids. We describe a method to compute the Helmholtz free energy of a crystalline solid by direct evaluation of the partition function. In the many-dimensional conformation space of all possible arrangements of N particles inside a periodic box, the energy landscape consists of localized islands corresponding to different solid phases. Calculating the partition function for a specific phase involves integrating over the corresponding island. Introducing a natural order parameter that quantifies the net displacement of particles from lattices sites, we write the partition function in terms of a one-dimensional integral along the order parameter, and evaluate this integral using umbrella sampling. We validate the method by computing free energies of both face-centered cubic (FCC) and hexagonal close-packed (HCP) hard sphere crystals with a precision of $10^{-5}k_BT$ per particle. In developing the numerical method, we find several scaling properties of crystalline solids in the thermodynamic limit. Using these scaling properties, we derive an explicit asymptotic formula for the free energy per particle in the thermodynamic limit. In addition, we describe several changes of coordinates that can be used to separate internal degrees of freedom from external, translational degrees of freedom. The second part of the dissertation focuses on engineering idealized physical devices that work as Maxwell's demon. We describe two autonomous mechanical devices that extract energy from a single heat bath and convert it into work, while writing information onto memory registers. Additionally, both devices can operate as Landauer's eraser, namely they can erase information from a memory register, while energy is dissipated into the heat bath. The phase diagrams and the efficiencies of the two models are solved and analyzed. These two models provide concrete physical illustrations of the thermodynamic consequences of information processing.

Irreversible thermodynamics of Poisson processes with reaction

A kinetic model is derived to study the successive movements of particles, described by a Poisson process, as well as their generation. The irreversible thermodynamics of this system is also studied from the kinetic model. This makes it possible to evaluate the differences between thermodynamical quantities computed exactly and up to second-order. Such differences determine the range of validity of the second-order approximation to extended irreversible thermodynamics

Predictions from information-theoretical models of nonequilibrium radiation

The radiation distribution function used by Domínguez and Jou [Phys. Rev. E 51, 158 (1995)] has been recently modified by Domínguez-Cascante and Faraudo [Phys. Rev. E 54, 6933 (1996)]. However, in these studies neither distribution was written in terms of directly measurable quantities. Here a solution to this problem is presented, and we also propose an experiment that may make it possible to determine the distribution function of nonequilibrium radiation experimentally. The results derived do not depend on a specific distribution function for the matter content of the system

Time-dependent couplings and crossover length scales in nonequilibrium surface roughening

We show that time-dependent couplings may lead to nontrivial scaling properties of the surface fluctuations of the asymptotic regime in nonequilibrium kinetic roughening models. Three typical situations are studied. In the case of a crossover between two different rough regimes, the time-dependent coupling may result in anomalous scaling for scales above the crossover length. In a different setting, for a crossover from a rough to either a flat or damping regime, the time-dependent crossover length may conspire to produce a rough surface, although the most relevant term tends to flatten the surface. In addition, our analysis sheds light into an existing debate in the problem of spontaneous imbibition, where time-dependent couplings naturally arise in theoretical models and experiments.

Finite-size effects in nonequilibrium correlation functions

We have shown that finite-size effects in the correlation functions away from equilibrium may be introduced through dimensionless numbers: the Nusselt numbers, accounting for both the nature of the boundaries and the size of the system. From an analysis based on fluctuating hydrodynamics, we conclude that the mean-square fluctuations satisfy scaling laws, since they depend only on the dimensionless numbers in addition to reduced variables. We focus on the case of diffusion modes and describe some physical situations in which finite-size effects may be relevant.

Thermostatistical description of small systems in nonequilibrium conditions: Energy conversion and harvesting

Hysteresis cycles are very important features of energy conversion and harvesting devices, such as batteries. The efficiency of these may be strongly affected by the physical size of the system. Here, we show that in systems which are small enough, the existence of physical boundaries which produce nonhomogeneities of the interaction potential gives rise to inflections and barriers in the associated free energy. This in turn brings on irreversible processes which can be triggered under suitable external conditions imposed by a heat bath. As an example, by controlling the temperature, the state of a small system may be impelled to oscillate between two different structural configurations or aggregation states avoiding equilibrium coexistence and therefore dissipating energy. This cyclical behavior associated with a hysteresis cycle may be prototypical of energy conversion, storage, or generating nanodevices, as exemplified by Li-ion insertion batteries.

Coarse-graining and thermodynamics in far-from-equilibrium systems

Lying at the core of statistical physics is the need to reduce the number of degrees of freedom in a system. Coarse-graining is a frequently-used procedure to bridge molecular modeling with experiments. In equilibrium systems, this task can be readily performed; however in systems outside equilibrium, a possible lack of equilibration of the eliminated degrees of freedom may lead to incomplete or even misleading descriptions. Here, we present some examples showing how an improper coarse-graining procedure may result in linear approaches to nonlinear processes, miscalculations of activation rates and violations of the fluctuation-dissipation theorem.

Critical behavior in nonequilibrium phase transitions

The nonequilibrium phase transitions occurring in a fast-ionic-conductor model and in a reaction-diffusion Ising model are studied by Monte Carlo finite-size scaling to reveal nonclassical critical behavior; our results are compared with those in related models.

Critical behavior in nonequilibrium phase transitions

The nonequilibrium phase transitions occurring in a fast-ionic-conductor model and in a reaction-diffusion Ising model are studied by Monte Carlo finite-size scaling to reveal nonclassical critical behavior; our results are compared with those in related models.

Entropy Production in Nonequilibrium Systems at Stationary States

We present a stochastic approach to nonequilibrium thermodynamics based on the expression of the entropy production rate advanced by Schnakenberg for systems described by a master equation. From the microscopic Schnakenberg expression we get the macroscopic bilinear form for the entropy production rate in terms of fluxes and forces. This is performed by placing the system in contact with two reservoirs with distinct sets of thermodynamic fields and by assuming an appropriate form for the transition rate. The approach is applied to an interacting lattice gas model in contact with two heat and particle reservoirs. On a square lattice, a continuous symmetry breaking phase transition takes place such that at the nonequilibrium ordered phase a heat flow sets in even when the temperatures of the reservoirs are the same. The entropy production rate is found to have a singularity at the critical point of the linear-logarithm type.

Non-equilibrium thermodynamics of harmonically trapped bosons

We apply the framework of non-equilibrium quantum thermodynamics to the physics of quenched small-size bosonic quantum gases in a harmonic trap. By studying the temporal behaviour of the Loschmidt echo and of the atomic density profile within the trap, which are informative of the non-equilibrium physics and the correlations among the particles, we establish a link with the statistics of (irreversible) work done on the system. This highlights interesting connections between the degree of inter-particle entanglement and the non-equilibrium thermodynamics of the system.