Extremal properties of distribution function and statistical operator in equilibrium and nonequilibrium thermodynamics

It is shown that the distribution function and the statistical operator, in the case that the considered system is close to the equilibrium state, can be received by the method relying upon minimizing the information gain, which is connected with the transition of the system from a nonequilibrium state to the equilibrium state. For the systems far from equilibrium the nonequilibrium distribution function or the nonequilibrium statistical operator can be derived using a variational principle based on Jaynes' maximum entropy formalism including memory effects.     

Statistical-Mechanical Foundations for a Generalized Thermodynamics of Dissipative Processes

A statistical-mechanical formalism for nonequilibrium systems, namely the nonequilibrium statistical operator method, provides microscopic foundations for a generalized thermodynamics of dissipative processes. This formalism is based on a unifying variational approach that is considered to be encompassed in Jaynes' Predictive Statistical Mechanics and principle of maximization of the statistical-informational entropy. Within the framework of the statistical thermodynamics that follows from the method, we demonstrate the existence of generalized forms of the theorem of minimum (informational) entropy production, the criterion for evolution, and the thermodynamic (in)stability criterion. The formalism is not restricted to local equilibrium but, in principle, to general conditions (its complete domain of validity is not yet fully determined). A H-theorem associated to the formalism is presented in the form of an increase of the informational entropy along the evolution of the system. Some of the results are illustrated in an application to the study of a model for a photoexcited direct-gap semiconductor.     

Regge theory and statistical mechanics

An interesting connection between the Regge theory of scattering, the Veneziano amplitude, the Lee–Yang theorems in statistical mechanics and nonextensive Renyi entropy is addressed. In this scheme the standard entropy and the Renyi entropy appear to be different limits of a unique mathematical object. This framework sheds light on the physical origin of nonextensivity. A non-trivial application to spin glass theory is shortly outlined.     

Statistical mechanics based on Renyi entropy

In this work we show that it is possible to obtain a generalized statistical mechanics (thermostatistics) based on Renyi entropy, to be maximized with adequate constraints. The equilibrium probability distribution thus obtained has a very interesting property. Indeed, it reminds us the statistical distribution proposed by Tsallis, known to conveniently describe a variety of phenomena in nonextensive systems. Moreover, some examples are worked out in order to illustrate the main features of the herein introduced formalism.     

The entropy flux and the evolution equations for the fluxes including nonlocal terms from the point of view of extended irreversible thermodynamics

The expression for the entropy flux is analysed from the point of view of irreversible thermodynamics. In connection with this problem the evolution equations for the heat flux and for the electric current density including nonlocal terms are derived and discussed. The relation for the entropy flux is compared with that obtained by the statistical nonequilibrium thermodynamics on the basis founded on a generalized Gibbs' ensemble method for nonequilibrium systems.     

Maximum Renyi entropy principle for systems with power-law Hamiltonians

The Renyi distribution ensuring the maximum of Renyi entropy is investigated for a particular case of a power-law Hamiltonian. Both Lagrange parameters alpha and beta can be eliminated. It is found that beta does not depend on a Renyi parameter q and can be expressed in terms of an exponent kappa of the power-law Hamiltonian and an average energy U. The Renyi entropy for the resulting Renyi distribution reaches its maximal value at q=1/(1+kappa) that can be considered as the most probable value of q when we have no additional information on the behavior of the stochastic process. The Renyi distribution for such q becomes a power-law distribution with the exponent -(kappa+1). When q=1/(1+kappa)+epsilon (0相似文献   

Entropy,Fisher Information and Variance with Frost-Musulin Potenial

This study presents the Shannon and Renyi information entropy for both position and momentum space and the Fisher information for the position-dependent mass Schrödinger equation with the Frost-Musulin potential. The analysis of the quantum mechanical probability has been obtained via the Fisher information. The variance information of this potential is equally computed. This controls both the chemical properties and physical properties of some of the molecular systems. We have observed the behaviour of the Shannon entropy. Renyi entropy, Fisher information and variance with the quantum number n respectively.     

Entropy maximum in a nonlinear system with the 1/<Emphasis Type="Italic">f</Emphasis> fluctuation spectrum

Analysis of the control and subordination is carried out for the system of nonlinear stochastic equations describing fluctuations with the 1/f spectrum and with the interaction of nonequilibrium phase transitions. It is shown that the control equation of the system has a distribution function that decreases upon an increase in the argument in the same way as the Gaussian distribution function. Therefore, this function can be used for determining the Gibbs-Shannon informational entropy. The local maximum of this entropy is determined, which corresponds to tuning of the stochastic equations to criticality and indicates the stability of fluctuations with the 1/f spectrum. The values of parameter q appearing in the definition of these entropies are determined from the condition that the coordinates of the Gibbs-Shannon entropy maximum coincide with the coordinates of the Tsallis entropy maximum and the Renyi entropy maximum for distribution functions with a power dependence.     

Steady state of photoexcited plasma in semiconductors

The nonequilibrium macroscopic state of a polar semiconductor under constant laser illumination is studied. The values of the relevant steady-state thermodynamic parameters are determined. They are the solutions of generalized transport equations derived using the nonequilibrium statistical operator method. Numerical results are obtained for the case of GaAs.     

Generating Functionals in Nonequilibrium Statistical Mechanics

The main ideas and methods of calculations within the framework of the generating functional technique are considered in a systematical way. The nonequilibrium generating functionals are defined as functional mappings of the nonequilibrium statistical operator and so appear to be dependent on a certain set of macroscopic variables describing the nonequilibrium state of the system. The boundary conditions and the differential equation of motion for the generating functionals are considered which result in an explicit expression for the nonequilibrium generating functionals in terms of the so-called coarse-grained generating functional being the functional mapping of the quasiequilibrium statistical operator. Various types of integral equations are derived for the generating functionals which are convenient to develop the perturbation theories with respect to either small interaction or small density of particles. The master equation for the coarse-grained generating functionals is obtained and its connection with the generalized kinetic equations for a set of macrovariables is shown. The derivation of the generalized kinetic equations for some physical systems (classical and quantum systems of interacting particles, the Kondo system) is treated in detail, with due regard for the polarization effects as well as the energy and momentum exchange between the colliding particles and the surrounding media.     

Improved signal processing to detect cancer by ultrasonic molecular imaging of targeted nanoparticles

In several investigations of molecular imaging of angiogenic neovasculature using a targeted contrast agent, Renyi entropy [I(f)(r)] and a limiting form of Renyi entropy (I(f,∞)) exhibited significantly more sensitivity to subtle changes in scattering architecture than energy-based methods. Many of these studies required the fitting of a cubic spline to backscattered waveforms prior to calculation of entropy [either I(f)(r) or I(f,∞)]. In this study, it is shown that the robustness of I(f,∞) may be improved by using a smoothing spline. Results are presented showing the impact of different smoothing parameters. In addition, if smoothing is preceded by low-pass filtering of the waveforms, further improvements may be obtained.     

Hamiltonian dynamics,nanosystems, and nonequilibrium statistical mechanics

An overview is given of recent advances in nonequilibrium statistical mechanics on the basis of the theory of Hamiltonian dynamical systems and in the perspective provided by the nanosciences. It is shown how the properties of relaxation toward a state of equilibrium can be derived from Liouville's equation for Hamiltonian dynamical systems. The relaxation rates can be conceived in terms of the so-called Pollicott–Ruelle resonances. In spatially extended systems, the transport coefficients can also be obtained from the Pollicott–Ruelle resonances. The Liouvillian eigenstates associated with these resonances are in general singular and present fractal properties. The singular character of the nonequilibrium states is shown to be at the origin of the positive entropy production of nonequilibrium thermodynamics. Furthermore, large-deviation dynamical relationships are obtained, which relate the transport properties to the characteristic quantities of the microscopic dynamics such as the Lyapunov exponents, the Kolmogorov–Sinai entropy per unit time, and the fractal dimensions. We show that these large-deviation dynamical relationships belong to the same family of formulas as the fluctuation theorem, as well as a new formula relating the entropy production to the difference between an entropy per unit time of Kolmogorov–Sinai type and a time-reversed entropy per unit time. The connections to the nonequilibrium work theorem and the transient fluctuation theorem are also discussed. Applications to nanosystems are described.     

Non-Markovian Quantum Kinetics and Conservation Laws

A link between memory effects in quantum kinetic equations and nonequilibrium correlations associated with the energy conservation is investigated. In order that the energy be conserved by an approximate collision integral, the one-particle distribution function and the mean interaction energy are treated as independent nonequilibrium state parameters. The density operator method is used to derive a kinetic equation in second-order non-Markovian Born approximation and an evolution equation for the nonequilibrium quasi-temperature which is thermodynamically conjugated to the mean interaction energy. The kinetic equation contains a correlation contribution which exactly cancels the collision term in thermal equilibrium and ensures the energy conservation in nonequilibrium states. Explicit expressions for the entropy production in the non-Markovian regime and the time-dependent correlation energy are obtained.     

General kinetic equation and irreversibility in statistical systems

General kinetic equation for statistical systems is presented. A kinetic equation with source that is fluctuation of physical values was obtained. A new statistical criterion of systems evolution was determined. Nonequilibrium statistical and variational derivations of general kinetic equations are considered. Evolution of nonequilibrium Boltzmann-Gibbs-Shannon entropy, Hamilton function and Hamilton function production are examined.     

非平衡涨落问题的微观唯象分析理论(Ⅰ)——一种新的广义不可逆热力学理论与热涨落中涨落—耗散表示式的非平衡修正

本文给出非平衡涨落问题的微观唯象分析理论——非平衡涨落的统计学描述理论。该理论的基础是广义非平衡熵与描述涨落几率的爱因斯坦表示式的推广。通过计算求得刚体热传导中比能与热通量的非平衡涨落的二阶矩。导出对热涨落的通用涨落-耗散表示式的非平衡修正,同时发现该修正相当于固体电介质中的光子热输运与金属中电子热输运的数值修正。     

Influence of environment and entropy production of a nonequilibrium open system

In the time-dependent projection operator formalism, the influence of environment upon a nonequilibrium open system is analyzed and an entropy equation is derived. The entropy production rate is given in terms of correlation functions of fluctuations of random forces and interacting random forces and cast into the Volterra equation formalism.     

A microscopic theory of nucleation. II

Using the nonequilibrium statistical operator obtained in the preceding paper of the authors [1], equations describing the kinetics of nucleation in a nonequilibrium medium are derived. A Fokker-Planck equation is found for embryo distribution functions in the number of particles, energy, momentum, and c.m. coordinates with additional random forces due to non equilibrium processes in the medium. Hydrodynamic equations are obtained for the medium with account of thermodynamic forces due to discontinuities of thermodynamic parameters at the interphase boundary. The symmetry of cross (interphase) kinetic coefficients is considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 44–52, May, 1978.     

Time Evolution in Macroscopic Systems. II. The Entropy

The concept of entropy in nonequilibrium macroscopic systems is investigated in the light of an extended equation of motion for the density matrix obtained in a previous study. It is found that a time-dependent information entropy can be defined unambiguously, but it is the time derivative or entropy production that governs ongoing processes in these systems. The differences in physical interpretation and thermodynamic role of entropy in equilibrium and nonequilibrium systems is emphasized and the observable aspects of entropy production are noted. A basis for nonequilibrium thermodynamics is also outlined.     

Unified projection operator formalism in nonequilibrium statistical mechanics

The method of projection operators, which plays an important role in the field of nonequilibrium statistical mechanics, has been established with the use of the Liouville-von Neumann equation for a density matrix to eliminate irrelevant information from a whole system. We formulate a unified and general projection operator method for dynamical variables. The main features of our formalism parallel those for the Liouville-von Neumann equation. (1) Two types of basic equations, time-convolution and time-convolutionless decompositions, are systematically obtained without specifying a projection operator. (2) Expansion formulas for both decompositions are also obtained. (3) Problems incorporating a time-dependent Liouville operator can be flexibly treated. We apply the formulas to problems in random frequency modulation and low field resonance. In conclusion, our formalism yields a more direct and easier means of determining the average time evolution of an operator than the one for the Liouville-von Neumann equation.