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.     

Quantum stochastic processes

We first define a class of processes which we call regular quantum Markov processes. We next prove some basic results concerning such processes. A method is given for constructing quantum Markov processes using transition amplitude kernels. Finally we show that the Feynman path integral formalism can be clarified by approximating it with a quantum stochastic process.     

Quantum stochastic processes

In order to describe rigorously certain measurement procedures, where observations of the arrival of quanta at a counter are made throughout an interval of time, it is necessary to introduce the concept of a quantum stochastic process. While fully quantum mechanical in nature, these have a great deal of similarity with classical stochastic processes and can be characterized by and constructed from their infinitesimal generators. The infinitestimal generators are naturally obtained from certain fields which we prove must be of the boson or fermion type.This work was supported by a National Science Foundation grant GP-7952X.     

Quantum stochastic processes II

We investigate properties of a class of quantum stochastic processes subject to a condition of irreducibility. These processes must be recurrent or transient and an equilibrium state can only exist in the former case. Every finite dimensional process is recurrent and it is possible to establish convergence in time to a unique equilibrium state. We study particularly the class of transition processes, which describe photon emissions of simple quantum mechanical systems in excited states.Work supported by U.S.A.F. contract number F 44620-67-C-0029.     

Quantum stochastic processes III

We construct an example of a quantum stochastic process with a non-zero, linear, time-independent source, for a massive scalar Boson field in four space-time dimensions. Also we study in detail a similar process with only a single degree of freedom.Work supported by A.F.O.S.R. contract no. F44620-67-C-0029     

Equilibrium thermodynamics and thermodynamic processes in nonlinear systems

A general method for the calculation of thermodynamic quasi-equilibrium processes by molecular dynamics (MD) simulation in canonical ensemble is developed. The method is suitable for classical systems with arbitrary interaction potentials. Though this MD method does not allow to calculate directly partition function and entropy, it is possible to calculate necessary partial derivatives which enter into the expressions for the full derivatives dT/dV and d β/dV for adiabitic and isobaric processes. Namely the solutions of these ordinary differential equations define the corresponding thermodynamic processes. The adiabatic process for the 1D Toda lattice is analyzed in details. The usage of the Toda potential allows to perform all analytical calculus up to accurate answers and to compare numerical and analytical results. Exact analytical expressions for the thermodynamics of 1D lattices with few types of nearest neighbor interactions are obtained as a necessary interim solutions. MD-simulation of quasi equilibrium processes in canonical ensemble demands the achievement of thermodynamic equilibrium, thus the thermalization kinetics is briefly discussed.     

Role of external flow and frame invariance in stochastic thermodynamics

For configurational changes of soft matter systems affected or caused by external hydrodynamic flow, we identify applied work, exchanged heat, and entropy change on the level of a single trajectory. These expressions guarantee invariance of stochastic thermodynamics under a change of frame of reference. As criterion for equilibrium versus nonequilibrium, zero versus nonzero applied work replaces detailed balance versus nonvanishing currents, since both latter criteria are shown to depend on the frame of reference. Our results are illustrated quantitatively by calculating the large deviation function for the entropy production of a dumbbell in shear flow.     

Multiplicative stochastic processes in statistical physics

For a large class of nonlinear stochastic processes with pure multiplicative fluctuations the corresponding time-dependent Fokker-Planck equation is solved exactly by means of analytic methods. We obtain a universal eigenvalue spectrum and the corresponding set of eigenfunctions.     

Quantum stochastic processes in curved space-time

A general scheme of quantum random processes is used for the simplest case of scalar fields in curved space-time. A covariant definition of a stochastic integral is introduced. ItÔ's formula is generalized and covariant Kolmogorov-Klein-Fock equations are obtained. The wave functions are expressed in the form of quantum expectations, and also in the form of path integrals.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 105–110, December, 1978.I should like to take this opportunity of expressing my deep thanks to Yu. S. Vladimirov for constant interest in the work.     

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: II. Kinetic potential and variational principles for nonlinear irreversible processes

On the basis of a complete system of fluctuation-dissipation relations, considered in the first part of this series, a variational principle for nonlinear irreversible processes is derived. According to this principle the virtual entropy production functional (analogous to the action in mechanics) has an absolute minimum meaning on the real trajectory of a system. The universal structure of the “kinetic potential” and the “lagrangian” of a system, each contain complete information about fluctuations of macrovariables. The connection of the lagrangian with the markovian kinetic operator of macrovariables is stated. Fundamental properties of dissipative potentials, reflecting microscopic reversibility, are considered. The derived variational principle can be applied to closed systems (the steady state of which is equilibrium) as well as to open ones (when external dynamic forces cause entropy flux through the system and put it into a steady non-equilibrium state). Canonical transformations of macrovariables are considered.     

Relationship between the thermodynamics of nonequilibrium state and the thermodynamics of nonequilibrium processes

The concept and equations of the thermodynamics of nonequilibrium state are compared with the postulates of linear thermodynamics of nonequilibrium processes. It is shown that all the four basic postulates can be obtained as consequences of the equations of the thermodynamics of nonequilibrium state. In the limiting case of small deviations from equilibrium, the thermodynamics of nonequilibrium state are reduced to linear thermodynamics of nonequilibrium processes.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 10–14, December, 1990.     

小系统的非平衡统计力学与随机热力学

热力学是一个古老的课题,古典热力学以宏观的具有大粒子数的系统为研究对象,自17世纪以来,科学家们构建了热力学的完备公理化体系。将热力学推广至小系统是近三十年来的研究前沿。文章介绍小系统的非平衡统计力学以及小系统的随机热力学。作为研究案例,利用时间依赖的谐振子势场控制单个粒子来构造随机热机的类卡诺循环,并发现该热机最大功率对应的效率等于1-,其中T_c和T_h分别对应于低温热库和高温热库的温度。     

Fermionic and supersymmetric stochastic processes

The fermionic Fock space is represented by the Wiener chaos. This identification allows one to define fermionic Brownian motion with a probability measure. In the underlying geometrical picture this Brownian motion evolves in the linear space of the generators of the Grassmann algebra which spans the Fock space. More general stochastic processes can be derived with the help of stochastic differential equations. The generalization to supersymmetric processes is based on the Wiener-Grassmann product of Le Jan, an algebraic structure which is adequate to investigate differential operators on Wiener spaces.     

Integrable models and stochastic processes

A general way for constructing square lattice systems with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These models give rise to series of integrable (stochastic) systems. As examples theAn-symmetric chain models and theSU(2)-invariant ladder models are investigated. Presented at the 10th Colloquium on Quantum Groups: “Quantum Groups and Intergrable Systems”, Prague, 21–23 June, 2001 SFB 256; BiBoS; CERFIM(Locarno); Acc. Arch.; USI(Mendriso)