Nonequilibrium thermodynamics at the microscale: Work relations and the second law

For macroscopic systems, the second law of thermodynamics establishes an inequality between the amount of work performed on a system in contact with a thermal reservoir, and the change in its free energy. For microscopic systems, this result must be considered statistically, as fluctuations around average behavior become substantial. In recent years it has become recognized that these fluctuations satisfy a number of strong and unexpected relations, which remain valid even when the system is driven far from equilibrium. We discuss these relations, and consider what they reveal about the second law of thermodynamics and the nature of irreversibility at the microscale.     

Two models for josephson oscillators

Two exactly soluble quantum mechanical models for tunneling oscillators are discussed. The first one is a closed weak coupling system, in which the current is nondissipative. In the second, open strong coupling system the oscillation occurs as a reservoir driven phase transition of the first kind far from thermal equilibrium.     

Thermal response in driven diffusive systems

Evaluating the linear response of a driven system to a change in environment temperature(s) is essential for understanding thermal properties of nonequilibrium systems. The system is kept in weak contact with possibly different fast relaxing mechanical, chemical or thermal equilibrium reservoirs. Modifying one of the temperatures creates both entropy fluxes and changes in dynamical activity. That is not unlike mechanical response of nonequilibrium systems but the extra difficulty for perturbation theory via path-integration is that for a Langevin dynamics temperature also affects the noise amplitude and not only the drift part. Using a discrete-time mesh adapted to the numerical integration one avoids that ultraviolet problem and we arrive at a fluctuation expression for its thermal susceptibility. The algorithm appears stable under taking even finer resolution.     

Nonequilibrium work relations: foundations and applications

When a macroscopic system in contact with a heat reservoir is driven away from equilibrium, the second law of thermodynamics places a strict bound on the amount of work performed on the system. With a microscopic system the situation is more subtle, as thermal fluctuations give rise to a statistical distribution of work values. In recent years it has been realized that such distributions encode surprisingly more information than one might expect from traditional thermodynamic arguments. I will discuss a number of exact results that relate equilibrium properties of the system, in particular free energy differences, to the fluctuations in the work performed during such a nonequilibrium process. I will describe the theoretical foundations of these relations, connections with irreversibility and the second law of thermodynamics, and potential experimental and computational applications.     

Thermostating by Deterministic Scattering: The Periodic Lorentz Gas

We present a novel mechanism for thermalizing a system of particles in equilibrium and nonequilibrium situations, based on specifically modeling energy transfer at the boundaries via a microscopic collision process. We apply our method to the periodic Lorentz gas, where a point particle moves diffusively through an ensemble of hard disks arranged on a triangular lattice. First, collision rules are defined for this system in thermal equilibrium. They determine the velocity of the moving particle such that the system is deterministic, time-reversible, and microcanonical. These collision rules can systematically be adapted to the case where one associates arbitrarily many degrees of freedom to the disk, which here acts as a boundary. Subsequently, the system is investigated in nonequilibrium situations by applying an external field. We show that in the limit where the disk is endowed by infinitely many degrees of freedom it acts as a thermal reservoir yielding a well-defined nonequilibrium steady state. The characteristic properties of this state, as obtained from computer simulations, are finally compared to those of the so-called Gaussian thermostated driven Lorentz gas.     

Non-Markovian stochastic processes: colored noise

We survey classical non-Markovian processes driven by thermal equilibrium or nonequilibrium (nonthermal) colored noise. Examples of colored noise are presented. For processes driven by thermal equilibrium noise, the fluctuation-dissipation relation holds. In consequence, the system has to be described by a generalized (integro-differential) Langevin equation with a restriction on the damping integral kernel: Its form depends on the correlation function of noise. For processes driven by nonequilibrium noise, there is no such a restriction: They are considered to be described by stochastic differential (Ito- or Langevin-type) equations with an independent noise term. For the latter, we review methods of analysis of one-dimensional systems driven by Ornstein-Uhlenbeck noise.     

Exergy flows as bases of constructal law

How do macroscopic systems react when imposed to external forces? A recent analysis of Carnot’s theorem has pointed out that the systems do not convert all the inflow energy to work and dissipation, but that some energy will be incorporated to internal processes. These exergy flows appear as the heat exchanged with a second thermal reservoir of a thermodynamic general engine. Flows of energy from surroundings are driven by internal forces to the systemic process. Also, when the system has evolved to reach a stationary state, internal flows manifest themselves as internal processes. Here, the entropy generation extrema theorem is used to prove how the macroscopic system will react upon forces that are imposed by the external surroundings. This provides the link between the entropy generation extrema approach and the constructal law.     

Thermal diffusivity and nuclear spin relaxation: a continuous wave free precession NMR study

Continuous wave free precession (CWFP) nuclear magnetic resonance is capable of yielding quantitative and easily obtainable information concerning the kinetics of processes that change the relaxation rates of the nuclear spins through the action of some external agent. In the present application, heat flow from a natural rubber sample to a liquid nitrogen thermal bath caused a large temperature gradient leading to a non-equilibrium temperature distribution. The ensuing local changes in the relaxation rates could be monitored by the decay of the CWFP signals and, from the decays, it was possible to ascertain the prevalence of a diffusive process and to obtain an average value for the thermal diffusivity.     

Coupled normal heat and matter transport in a simple model system.

We introduce the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system is a Lorentz gas with fixed freely rotating circular scatterers which scatter point particles via perfectly rough collisions. Upon imposing either a temperature gradient and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds. Transport in this system is normal in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are nontrivially coupled, satisfying Onsager's reciprocity relations.     

热储周围岩石热补偿对增强型地热系统采热过程的影响

采用自行开发的增强型地热系统(EGS)地下热流动过程三维动态模拟软件,模拟不同地质条件下EGS的长期运行过程,分析热储周围岩体的热补偿对产热温度以及热储内岩石、流体温度演化的影响.该数值模型视热储为等效多孔介质,采用两个能量方程分别描述流体和岩石的温度场,深入探究岩石与循环流体之间的换热过程.研究发现,热储周围岩体的热补偿作用与热储内流场形态强烈相关,且并不总是提高EGS的生产温度.在深度方向上有较大的优势流动的热储中,热补偿作用在EGS运行早期甚至会降低采出流体的温度.随着EGS的运行,热储温度持续降低,热补偿将对热能开采的影响将逐渐转向正面,对生产流体温度的提高效果增强.     

Chemical and Mechanical Aspect of Entropy-Exergy Relationship

The present research focuses the chemical aspect of entropy and exergy properties. This research represents the complement of a previous treatise already published and constitutes a set of concepts and definitions relating to the entropy–exergy relationship overarching thermal, chemical and mechanical aspects. The extended perspective here proposed aims at embracing physical and chemical disciplines, describing macroscopic or microscopic systems characterized in the domain of industrial engineering and biotechnologies. The definition of chemical exergy, based on the Carnot chemical cycle, is complementary to the definition of thermal exergy expressed by means of the Carnot thermal cycle. These properties further prove that the mechanical exergy is an additional contribution to the generalized exergy to be accounted for in any equilibrium or non-equilibrium phenomena. The objective is to evaluate all interactions between the internal system and external environment, as well as performances in energy transduction processes.     

From space-time chaos to stochastic thermodynamics

We summarize the results of several experiments that show the evolution of some scientific interests and goals of the statistical and nonlinear physics community in the last 40 years. Specifically, we present how the ideas of extending concepts of equilibrium statistical physics to out-of-equilibrium physics have been developed to characterize various phenomena such as, for example, transition to space-time chaos and glass aging. We then discuss the applications of this out-of-equilibrium thermodynamics to microsystems driven out of equilibrium either by external forces or by temperature gradients. We show that in these systems thermal fluctuations play a role and that all thermodynamics quantities, such as work, heat, and entropy fluctuate. We recall general concepts such as fluctuation theorems and fluctuation dissipation relations used to characterize the statistical properties of these small systems. We describe experiments where all these concepts have been applied and tested with high accuracy. Finally, we show how these theoretical concepts and the experiments allowed us to improve our knowledge on the connection between information and thermodynamics.     

Asymmetric response of a jammed plastic bead raft

The successful development of an effective temperature would be an important step in the application of statistical mechanics principles to systems driven far from equilibrium. One direction that has shown promise is the use of fluctuation-dissipation relations. However, driven systems break time-reversal symmetry, and understanding the implications of this for fluctuation-dissipation relations is a critical step in developing effective temperatures. Here we study the response function in a driven system of plastic beads as a function of the density in order to elucidate the generality of the use of fluctuation-dissipation relations. We find that even when a linear response is observed, the time scale of the response is dependent on the direction of the applied stress.     

Heat flow and efficiency in a microscopic engine

We study the energetics of a thermal motor driven by temperature differences, which consists of a Brownian particle moving in a sawtooth potential with an external load where the viscous medium is periodically in contact with hot and cold heat reservoir along space coordinate. The motor can work as a heat engine or a refrigerator under different conditions. The heat flow via both potential and kinetic energy is considered. The former is reversible when the engine works quasistatically and the latter is always irreversible. The efficiency of the heat engine can never approach Carnot efficiency.     

COMMENTS ON “THERMODYNAMIC UNCERTAINTY RELATIONS” BY J. UFFINK AND J. VAN LITH

Fundamental errors in the Uffink and van Lith paper [Found. Phys. 29, 655-92 (1999)] are highlighted. Contrary to their claim, thermodynamic uncertainty relations are derived from the second law. To first order in the energy difference, the increase in entropy when two systems at slightly different temperatures are placed in thermal contact is equal to the product of the second subsystems multiplied by the change in energy of the first subsystem. This expression for the increase in entropy has the same range of validity as the thermodynamic uncertainty relations and the Fisher information, i.e., they are second order quantities. The dispersion in the inverse temperature of either subsystem cannot be less than the inverse of the Fisher information, or the inverse of the dispersion in energy of the composite system at the lowest possible equilibrium temperature, where the equality applies strictly to reversible processes and the inequality to irreversible ones.     

Thermalizing quantum machines: dissipation and entanglement

We study the relaxation of a quantum system towards the thermal equilibrium using tools developed within the context of quantum information theory. We consider a model in which the system is a qubit, and reaches equilibrium after several successive two-qubit interactions (thermalizing machines) with qubits of a reservoir. We characterize completely the family of thermalizing machines. The model shows a tight link between dissipation, fluctuations, and the maximal entanglement that can be generated by the machines. The interplay of quantum and classical information processes that give rise to practical irreversibility is discussed.     

温库边界对布朗热机性能的影响

研究了单势垒锯齿势中,布朗粒子在外力和空间周期温度场作用下构成的布朗热机的热力学性能.考虑布朗粒子动能变化以及高、低温库之间热漏引起的热流.用Smoluchowski方程描述粒子在黏性介质中的动力学特性,推导出高、低温库的热流以及热机功率和效率的解析表达式.通过数值计算分析势垒高度、外力和温库边界对热机性能的影响.研究表明:由于动能变化和热漏引起的不可逆热流的存在,布朗热机为不可逆热机,热机的功率效率特性为一闭合的关系曲线;势垒边界与温库边界重合时,热机的功率达到最大值;通过改变温库边界的位置,可以在一定范围内提高热机的效率,但同时减小了热机的输出功率.     

Transport Properties of a Modified Lorentz Gas

We present a detailed study of the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system under consideration is a Lorentz gas with fixed freely-rotating circular scatterers interacting with point particles via perfectly rough collisions. Upon imposing a temperature and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds for low values of the imposed gradients. Transport in this system is normal, in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are non-trivially coupled, satisfying Onsager's reciprocity relations to within numerical accuracy as well as the Green–Kubo relations. We further show numerically that an applied electric field causes the same currents as the corresponding chemical potential gradient in first order of the applied field. Puzzling discrepancies in higher order effects (Joule heating) are also observed. Finally, the role of entropy production in this purely Hamiltonian system is shortly discussed.     

海森堡XYZ自旋链系统的热纠缠与局域量子不确定性研究

研究了热平衡温度,自旋交换相互作用,Dzyaloshinskii-Moriya(DM)相互作用及外加非一致性磁场对两比特海森堡XYZ自旋链量子系统的热纠缠与局域量子不确定度的影响,对比分析了并发度量子纠缠与局域量子不确定度描述自旋链系统量子关联的差别.结果表明自旋链系统的量子纠缠在热平衡温度,DM相互作用及外加磁场的非一致性参数的变化情况下均会出现纠缠突然死亡的再生现象,而自旋链系统的局域量子不确定度随着这些参数呈连续变化现象.并且,自旋交换相互作用,DM相互作用及外加横向磁场作用强度较小时,他们的变化对自旋链系统的量子纠缠与局域量子不确定度的影响有着明显的差别.     

Phonon induced tunneling of ions in solids

A quantum mechanical analysis is given of the change of position or orientation of an atom, ion or molecule in a crystal as it occurs e.g. in the processes of diffusion or hindered rotation. One-phonon processes, Raman processes and indirect processes are discussed. The limits of applicability of the classical rate theory are given. Specifically, the analysis is applied to the reorientation of the O₂- center in alkali halides under the influence of external mechanical stresses at low temperatures, which has been investigated experimentally byKänzig. The measured dependence of the reorientation time upon temperature and applied stress can be explained by one-phonon processes. Random internal strains in the crystal are shown to play an important part. As further application of the theory the proton motions in ice and in iron are elucidated. Finally, an estimate is given of the effect of direct processes involving imperfections on the thermal conductivity of alkali halides.