On the problem of the metastable region at morphological instability

The study deals with numerical analysis of the morphological stability of a growing round particle with respect to harmonic perturbations of an arbitrary amplitude. Various growth regimes (from diffusion to kinetic-limited) are considered. It is found that the critical size of the particle stability decreases as the perturbation amplitude increases and tends to the value, which was determined analytically elsewhere using the maximum entropy production principle. This result is a crucial argument in support of the hypothesis that the entropy production can be used for analysis of a nonequilibrium phase transitions similarly to thermodynamic potentials in the case of equilibrium phase transitions.     

Thermodynamics of an interfacial fluid membrane

We discuss a closed system of field equations for a semipermeable membrane which has particle and heat exchange with its surroundings. In this case we consider a surface with an arbitrary shape for specific quantities and mechanical properties. A representation of the constitutive equations follows from the principle of material objectivity in space as well as on surfaces. The constitutive equations can be restricted by an entropy principle. We present both the Gibbs equation and the entropy flux. Furthermore, we obtain the surface stress and the chemical potential in terms of the specific free energy of the membrane. Both the heat flux and the particle flux normal to the membrane depend on the mean curvature and the friction between the particle across the membrane. The interaction tangential to the interface is dependent up on gradients of the surface stress as well as the chemical potential of the interface.     

A method to calculate Boltzmann entropy

A method is proposed to calculate the Boltzmann non equilibrium entropy as a Taylor series expansion in terms of the successive moments of the velocity distribution function. As a first application, the entropy of the BKW solution of the Boltzmann equation is calculated for both even and odd dimensions. The properties of the entropy of the Tjon Wu modeld=2) are studied and a quantitative condition is derived, showing that the McKean conjecture is incorrect. As a second application of the method, the entropy of one of the solutions of the very hard particle model for the Boltzmann equation is also derived.     

On a kinetic justification of the generalized nonequilibrium thermodynamics of multicomponent systems

The problem of the kinetic justification of the generalized thermodynamics of nonequilibrium processes using the method of moments for solving the kinetic equation for a multicomponent gas mixture is examined. Generalized expressions are obtained for the entropy density, entropy flux density, and entropy production as functions of an arbitrary number of state variables (moments of the distribution function). Different variants of writing the relations between fluxes and thermodynamic forces are considered, which correspond to the Onsager version for spatially homogeneous systems and, in a more general case, lead to the generalized thermodynamic forces of a complicated form, including derivatives of the fluxes with respect to time and spatial coordinates. Some consequences and new physical effects, following from the obtained equations, are analyzed. It is shown that a transition from results of the method of moments to expressions for the entropy production and the corresponding phenomenological relations of the generalized nonequilibrium thermodynamics is possible on the level of a linearized Barnett approximation of the Chapman–Enskog method.     

色噪声驱动的双奇异随机系统的熵流与熵产生

讨论一类 (非平衡约束下) 高斯色噪声驱动的双奇异随机系统对应的Fokker-Planck方程，结合Shannon信息熵定义给出此类系统的熵流、熵产生随时间演化的精确表达式.分析(非平衡约束下)所引入的奇异性强度参数、噪声相关时间与耗散参数的相互作用以及对熵流、熵产生的显著影响. 关键词： 信息熵 熵流与熵产生 双奇异随机系统 高斯色噪声     

A decomposed equation for local entropy and entropy production in volume-preserving coarse-grained systems

In this study an equation for the local entropy is derived based on the formulation of a master equation and is applied to volume-preserving maps. The equation consists of the following terms: unsteady, convection, diffusion, probability-weighted phase space volume expansion rate, nonnegative entropy production, and residuals. The decomposition makes it possible to evaluate entropy production in terms of microscopic dynamics and is expected to be applicable to many coarse-grained systems on the phase space. When it is applied to two volume-preserving multibaker chain systems it is confirmed that the summation of the nonnegative entropy production on each site numerically coincides with the entropy production introduced by Gilbert et al. [T. Gilbert, J.R. Dorfman, P. Gaspard, Entropy production, fractals, and relaxation to equilibrium, Phys. Rev. Lett. 85 (2000) 1606-1609] and the phenomenological expression both in nonequilibrium steady and unsteady states. The coincidence is brought about by the fact that the residual terms vanish in the thermodynamic limit when they are integrated on each site. It follows that the entropy production is dominated by the nonnegative entropy production term and becomes positive in nonequilibrium states.     

A Scheme for Information Erasure in a Double-Well Potential

We design an experimental scheme to realize one-bit information erasure and restoring processes by considering an overdamped colloidal particle in a double-well optical trap, which is added by a controllable laser tweezer. Using the Monte Carlo method, we simulate numerically the Langevin equation to calculate the mean work spent during the entire process and validate the entropy production fluctuation theory. Our result shows that the distribution of entropy production becomes narrow with increasing temperature and becomes stationary, represents the diminishing extent of irreversibility.     

Charged Brownian particles: Kramers and Smoluchowski equations and the hydrothermodynamical picture

We consider a charged Brownian gas under the influence of external and non-uniform electric, magnetic and mechanical fields, immersed in a non-uniform bath temperature. With the collision time as an expansion parameter, we study the solution to the associated Kramers equation, including a linear reactive term. To the first order we obtain the asymptotic (overdamped) regime, governed by transport equations, namely: for the particle density, a Smoluchowski-reactive like equation; for the particle’s momentum density, a generalized Ohm’s-like equation; and for the particle’s energy density, a Maxwell-Cattaneo-like equation. Defining a nonequilibrium temperature as the mean kinetic energy density, and introducing Boltzmann’s entropy density via the one particle distribution function, we present a complete thermohydrodynamical picture for a charged Brownian gas. We probe the validity of the local equilibrium approximation, Onsager relations, variational principles associated to the entropy production, and apply our results to: carrier transport in semiconductors, hot carriers and Brownian motors. Finally, we outline a method to incorporate non-linear reactive kinetics and a mean field approach to interacting Brownian particles.     

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.     

非平衡统计信息理论

阐述了以表述信息演化规律的信息（熵）演化方程为核心的非平 衡统计信息理论.推导出了 Shannon信息（熵）的非线性演化方程，引入了统计物理信息并 推导出了它的非线性演化方程.这两种信息（熵）演化方程一致表明：统计信息（熵）密度 随时间的变化率是由其在坐标空间（和态变量空间）的漂移、扩散和减损（产生）三者引起 的.由此方程出发，给出了统计信息减损率和统计熵产生率的简明公式、漂移信息流和扩散 信息流的表达式，证明了非平衡系统内的统计信息减损（或增加）率等于它的统计熵产生（ 或减少）率、信息扩散与信息减损同时 关键词： 统计信息（熵）演化方程 统计信息减损率 统计熵产 生率 信息（熵）流 信息（熵）扩散 动态互信息     

非平衡热力学中传热过程熵产表达式的修正

熵产是非平衡热力学中的核心物理量,传统上表示为广义力(驱动力)与广义流的乘积.这种表达存在两方面缺陷:一是广义力与广义流的拆分具有任意性;更重要的是,以其计算热波传递时熵产可以为负值,从而违反热力学第二定律.本文基于热质理论分析表明,传热过程的熵产实质上是由热质流体的热质能耗散引起的,所以熵产中的力不是驱动力而是阻力,并且具有力的量纲.由此提出的熵产修正表达式,不仅在计算热波传递过程中熵产恒为正值,与扩展不可逆热力学中的熵产表达式一致,而且不存在力和流拆分的任意性.     

Entropy Production in Exactly Solvable Systems

The rate of entropy production by a stochastic process quantifies how far it is from thermodynamic equilibrium. Equivalently, entropy production captures the degree to which global detailed balance and time-reversal symmetry are broken. Despite abundant references to entropy production in the literature and its many applications in the study of non-equilibrium stochastic particle systems, a comprehensive list of typical examples illustrating the fundamentals of entropy production is lacking. Here, we present a brief, self-contained review of entropy production and calculate it from first principles in a catalogue of exactly solvable setups, encompassing both discrete- and continuous-state Markov processes, as well as single- and multiple-particle systems. The examples covered in this work provide a stepping stone for further studies on entropy production of more complex systems, such as many-particle active matter, as well as a benchmark for the development of alternative mathematical formalisms.     

A special class of solutions of the Schrödinger equation for a free particle

The fundamental solution of the Schrödinger equation for a free particle is modified by the inclusion of an arbitrary scalar and an arbitrary vector, both imaginary. This gives a field free from singularities. By choosing the scalar small and the vector large, one obtains a model of a wavepacket which moves fast and remains concentrated over a long range.     

Hamiltonian and Lagrangian for the Trajectory of the Empirical Distribution and the Empirical Measure of Markov Processes

We compute the Hamiltonian and Lagrangian associated to the large deviations of the trajectory of the empirical distribution for independent Markov processes, and of the empirical measure for translation invariant interacting Markov processes. We treat both the case of jump processes (continuous-time Markov chains and interacting particle systems) as well as diffusion processes. For diffusion processes, the Lagrangian is a quadratic form of the deviation of the trajectory from the solution of the Kolmogorov forward equation. In all cases, the Lagrangian can be interpreted as a relative entropy or relative entropy density per unit time.     

Zur Theorie der Teilchenbewegung in zeitabhängigen elektromagnetischen Feldern

The Langevin equation – i.e. the equation of motion for a charged particle including a collision term proportional to the particle velocity – is solved for arbitrary time-dependent electric and magnetic fields by a new general method. Instead of the usual ansatz: particle velocity = cyclotron velocity + drift velocity the method given makes the ansatz: particle velocity = tensor = cyclotron velocity. The unknown tensor obeys a simple differential equation of the first order which can be generally solved at once. This method is a modification of the variation of constants method for inhomogeneous differential equations. The electromagnetic fields considered must be spatially homogeneous; for (weakly) inhomogeneous fields an iteration procedure of Pytte (1962) may be applied. Some examples are discussed shortly. The Langevin equation treated is completely equivalent to the equation of motion in a magnetohydrodynamic one-fluid theory.     

Exact kinetic transport equation solutions in the particle propagation theory in the scattering medium

Charged particle kinetics in an inhomogeneous medium (stochastic magnetic field) is investigated. Exact analytic expressions for Green function of kinetic equation in relaxation-time approximation are derived in one and three dimensions with arbitrary particle absorption. We separately consider the case of isotropic particle injection as well as the case of unidirectional instantaneous particle injection. The new way of solution makes it possible to get off any Cauchy-Principal Value integrals in some solutions which arise in the inverse Fourier-Laplace transform. Weak scattering regime and diffusion approximation is considered, and particle density is derived in three dimensions and arbitrary particle source.     

A Photon Force and Flow for Dissipative Structuring: Application to Pigments,Plants and Ecosystems

Through a modern derivation of Planck’s formula for the entropy of an arbitrary beam of photons, we derive a general expression for entropy production due to the irreversible process of the absorption of an arbitrary incident photon spectrum in material and its dissipation into an infrared-shifted grey-body emitted spectrum, with the rest being reflected or transmitted. Employing the framework of Classical Irreversible Thermodynamic theory, we define the generalized thermodynamic flow as the flow of photons from the incident beam into the material and the generalized thermodynamic force is, then, the entropy production divided by the photon flow, which is the entropy production per unit photon at a given wavelength. We compare the entropy production of different inorganic and organic materials (water, desert, leaves and forests) under sunlight and show that organic materials are the greater entropy-producing materials. Intriguingly, plant and phytoplankton pigments (including chlorophyll) reach peak absorption exactly where entropy production through photon dissipation is maximal for our solar spectrum 430<λ<550 nm, while photosynthetic efficiency is maximal between 600 and 700 nm. These results suggest that the evolution of pigments, plants and ecosystems has been towards optimizing entropy production, rather than photosynthesis. We propose using the wavelength dependence of global entropy production as a biosignature for discovering life on planets of other stars.     

Microscopic irreversibility in the neutral kaon system and the thermodynamical arrow of time I. CPT symmetric case

The master equation for the reduced density matrix of the neutral kaon system interacting with a (low temperature) boson gas is derived. The equation is transformed under time reversal (assuming CPT conservation) and the equations of motion in the direct and inverted time schemes are compared. The equations of motion are solved, in a perturbation method, and the explicit approach to thermodynamical equilibrium is described. The entropy production function for the kaonic decaying system is defined and calculated. It is shown that although the equilibrium state and the total entropy change during the approach to it are the same in both time schemes, the details of the entropy time dependence of a coherent initial state are different, thus suggesting a relation between microscopic and macroscopic arrows of time.     

An optimal principle in fluid-structure interaction

We study the steady terminal orientation of a fore-aft symmetric body as it settles in a viscous fluid. An optimal principle for the settling behavior is discussed based upon entropy production in the system, both in the Stokes limit and the case of near equilibrium states when inertial effects emerge. We show that in the Stokes limit, the entropy production in the system is zero allowing any possible terminal orientation while in the presence of inertia, the particle assumes a horizontal position which coincides with the state of maximum entropy production. Our results are seen to agree well with experimental observations.     

Entropy Production Rate of Non-equilibrium Systems from the Fokker-Planck Equation

The entropy production rate of non-equilibrium systems is studied via the Fokker-Planck equation. This approach, based on the entropy production rate equation given by Schnakenberg from a master equation, requires information on the transition rate of the system under study. We obtain the transition rate from the conditional probability extracted from the Fokker-Planck equation and then derive a new and more operable expression for the entropy production rate. A few examples are presented as applications of our approach.