A minimal model with stochastically broken reciprocity

The method of classical statistical mechanics is applied to an inviscid truncated model system of two-dimensional turbulent shear flow. The idea of canonical equilibrium distribution is extended to treat a time-dependent Liouville equation governing the evolution of probability distribution function in the phase space. Results of numerical simulations supporting the theoretical conjectures based on the canonical distribution are shown.

A numerical solver for a nonlinear Fokker–Planck equation representation of neuronal network dynamics

Dynamical stochastic processes of returns in financial markets

The major goal of this work is to provide a means to characterize the final structure of a metal that has solidified from a melt. The thermally controlled solidification of a binary alloy, nucleated at isolated sites, is described by the evolution of a probability distribution function (PDF). The relevant equation required for propagating the PDF is developed with variables for grain size and distance to nearest neighbor. The phenomena of nucleation, growth, and impingement of the grains are discussed, and used as the basis for developing rate equations that evolve the PDF. The complementary equations describing global heat and solute transfer are discussed, and coupled with the microstructure evolution equations for grain growth and PDF evolution. The full set of equations is solved numerically and results are compared with experimental data for the plutonium 1 weight percent gallium system. The three principal results of this work are: (1) The formulation of transient evolution equations for the PDF description of nucleation, growth, and impingement of a distribution of grain sizes and locations; (2) Solution of the equations to give a correlation for final average grain size as a function of material parameters, nucleation site density, and cooling rate; and (3) Solution of the equations for final distribution of grain size as a result of the initial random spatial distribution of nucleation sites.

In laboratory fusion devices radio frequency electromagnetic waves are routinely used for heating plasmas and for controlling current profiles. The evolution of particle distribution function in the presence of electromagnetic waves is derived from fundamental equations using the action-angle variables of the dynamical Hamiltonian. Unlike conventional quasilinear theories (QLTs), the distribution function is evolved concurrently with the particle motion. Since the particle dynamics is time reversal invariant, the master equation for the evolution of the distribution function is also time reversal invariant. A sequential averaging of the master equation over the angles leads to a hierarchy of diffusion equations. The diffusion operator in the equation obtained after averaging over all angles is time dependent, in direct contrast to time independent diffusion operator in QLTs. The evolution of the distribution function with time-dependent diffusion operator is markedly different from quasilinear evolution and is illustrated for current drive by a spectrum of coherent electrostatic waves. A proper description of wave–particle interactions is important for fusion plasmas since the velocity space gradients of the distribution function decisively affect collisional relaxation and the associated transport processes.

We review the current state of a fundamental problem of rigorous derivation of transport processes in classical statistical mechanics from classical mechanics. Such derivations for diffusion and momentum transport (viscosities) were obtained for minimal models of these processes involving one and two particles respectively. However, a minimal model which demonstrates heat conductivity contains three particles. Its rigorous analysis is currently out of reach for existing mathematical techniques. The gas of localized balls is widely accepted as a basis for a simplest model for derivation of Fourier’s law. We suggest a modification of the localized balls gas and argue that this gas of localized activated balls is a good candidate to rigorously prove Fourier’s law. In particular, hyperbolicity is derived for a reduced version of this model.

Entropy production is the key to the second law of thermodynamics, and it is well defined by considering a joint unitary evolution of a system $S$ and a thermal environment $E$. However, due to the diversity of the initial state and Hamiltonian of the system and environment, it is hard to evaluate the characterisation of entropy production. In the present work, we propose that the evolution of $S$ and $E$ can be solved non-perturbatively in the framework of Gaussian quantum mechanics (GQM). We study the entropy production and correlation spreading in the interaction between Unruh-DeWitt-like particle detector and thermal baths, where the particle detector is set to be a harmonic oscillator and the thermal baths are made of interacting and noninteracting Gaussian states. We can observe that the entropy production implies quantum recurrence and shows periodicity. In the case of interacting bath, the correlation propagates in a periodic system and leads to a revival of the initial state. Our analysis can be extended to any other models in the framework of GQM, and it may also shed some light on the AdS/CFT correspondence.

A systematic approximation method for time evolutions of distribution functions of deterministic dynamical systems is developed when the distribution functions are not far from Gaussian. For simplicity and definiteness, the procedure of the method is shown by taking a simple dynamical system, that is, one-dimensional autonomous nonlinear oscillator. An essential point of the method is to replace an original initial value problem of the evolution of a distribution function by an appropriate initial value problem of the evolution of a Gaussian distribution function by using the Kullback-Leibler's information. Basic properties of an initial value problem posed by the method are also studied.

The evolution of a system of electrons with a given initial distribution in an external magnetic field is considered. An equation describing the evolution of the electron distribution function in a uniform magnetic field is derived for the case of arbitrarily relativistic electrons, and an exact solution to this equation is found. Asymptotics of this solution corresponding to the cases of synchrotron radiation and relativistic dipole radiation are calculated, and the evolution of the radiation spectra for these limiting cases is analyzed. The curvature of the magnetic field lines is taken into account phenomenologically, which demonstrates the presence of an exponential dependence in the case of synchrotron radiation.

Evaluating the energy loss of an electrically (color) charged particle crossing a high-temperature QED (QCD) plasma at its thermal equilibrium is studied. The average energy loss depends on the particle characteristics, plasma parameters, and QED (QCD) coupling constant alpha (alpha s). All processes through which the energy of a particle changes can be categorized into two main mechanisms: elastic collisions and radiation through bremsstrahlung. We have introduced the final results of collisional and radiation energy loss for an electrically charged particle in a QED plasma, as well as a quark in a QCD plasma. The suppression due to radiation is presented using the Landau-Pomeranchuk-Migdal effect. Time evolution of particle distribution functions has been evaluated numerically through the Fokker-Planck equation. We have calculated the drag and diffusion coefficients using the collisional and radiation energy loss definitions. outcomes of different presented relations are different. We have compared differences and similarities in evolution of distribution functions.

Ilya Prigogine

Time-dependent calculations are presented which compare the dynamics of quantum and classical one-dimensional triatomic systems undergoing intramolecular vibrational energy transfer and dissociation. The purpose of these calculations is to determine whether statistical dissociative behavior in classical systems implies similar behavior in the analogous quantum systems and to test for the presence of quantum mechanical effects that reduce the tendency of the systems to decompose statistically. The intramolecular vibrational energy transfer is monitored by computing the probability for the systems to remain in their initial, coarsely grained states, and the dissociation is followed by calculating the time-dependent decomposition probability and the product distribution. The classical calculations are performed by a version of the quasiclassical technique while the quantum calculations are carried out by an R-matrix method. The results show that the two forms of dynamics usually result in similar intramolecular evolution and unimolecular decay. Since the behavior of the classical systems is statistical in a well-defined sense, it is argued that the behavior of the quantum mechanical systems can likewise be labeled as statistical in these typical cases. Important exceptions to the generally good quantum-classical agreement occur, however, when the systems are prepared with high energy in a dissociable bond and low energy in the other bond. In such cases, the quantum behavior differs significantly from the classical behavior; the quantum dynamics of decomposition is nonstatistical even though the classical dynamics is statistical. It is found that the ‘‘quantum trapping’’ states which lead to the nonstatistical quantum mechanical behavior are associated with narrow Feshbach resonances which accumulate in certain specific energy regions. It is further concluded that these states occupy a significant proportion of the classical phase space available to molecular complexes with energy below the first vibrational threshold.

In this paper, we study the statistical evolution of the large-scale structure (LSS), focusing on the joint probability distribution function (PDF) of the coarse-grained cosmic field and its role in constructing effective dynamics.As the most comprehensive statistics, this PDF encodes all cosmological information of large-scale modes, therefore, could serve as the basis in the LSS modelling. Following the so-called PDF-based method from turbulence, we write down this PDF's evolution equation, which describes the probability conservation.We show that this conservation equation's characteristic curves follow the same PDF history and could be considered as an effective dynamics of the coarse-grained field.Unlike the EFT of LSS, which conceptually would work at both realization and statistics level, this effective dynamics is valid only statistically. However, this `statistical equivalence' also provides valuable insight into scale interactions at the statistical level. It also enables predicting a wide variety of statistics beyond the typical N-point polyspectra, including, e.g. topologies, density PDF and non-linear covariance matrices etc. Our formula expresses the small-scale effect as the ensemble average of their interactions conditional on the large-scale modes. This suggests an interesting way to measure effective terms directly from simulation. By applying the Gram-Charlier expansion, we demonstrate a different structure of these effective terms.This formalism is a natural framework for discussing the evolution of statistical properties of large-scale modes, and provides an alternative view for understanding the relationship between general effective dynamics and standard perturbation theory.

When a quantum system is coupled to two thermal baths at different temperatures, the temperature gradient as a thermodynamic force causes stationary heat flow within the system. We present a theoretical model to study how quantum features of squeezed thermal reservoirs affect the classical formulation of entropy production in terms of generalized forces and flows. Applying the quantum phase-space method to calculate Wigner entropy production helps us identify heat and squeezing fluxes and their corresponding generalized forces in terms of input-noise correlations of both reservoirs. Our study highlights the essential role played by the correlation of the input-noise operators of the thermal squeezed bath, which are asymmetric (squeezed and unsqueezed) and far from thermal equilibrium. Using the framework for an optomechanical system, we find the regime in which heat flows from the cold squeezed thermal bath to the hot thermal bath without violating the second law of thermodynamics and the regime in which entropy production is purely a quantum-mechanical effect. The results can be useful for improving the performance of Gaussian heat engines operating with squeezed thermal reservoirs.

We explored the Cauchy problem for the evolution of the charge density distribution function for a spherically symmetric system with nonzero initial conditions. In our model, the evolution of the charge density distribution function is simulated for the case of a non-uniform charged sphere. The initial speed of the system is nonzero. The solution breaks down into two components: the first one describes the system’s motion as a whole and the second describes the process of the evolution of the charge density function under the influence of its own electric field in the center-of-mass system. In this paper we considered the characteristic features of the implementation of a difference scheme for numerical simulation. We also illustrate the process of “scattering” of a moving charged system under the influence of its own electric field on the basis of the solution of the Cauchy problem for vector functions of the electric field and vector velocity field of a charged medium.