Nonequilibrium Fluctuations of a Brownian Particle in a Medium with Production of Entropy

The study statistically describes Brownian motion in a locally nonequilibrium medium, taking into account the production of entropy, and proposes to describe the nonequilibrium fluctuations of the velocity of a Brownian particle using a linear integro-differential equation. The characteristic functions of fluctuations of the Brownian particle velocity are obtained, which make it possible to carry out a complete statistical description of Brownian motion in a medium with the production of entropy. Findings of research show that the variance of these fluctuations increases with time according to the logarithmic law. The correlation function of fluctuations of the Brownian particle velocity is calculated and it is shown that it consists of two terms. The first term, which has a power-law dependence, describes equilibrium fluctuations, and the second, which has a logarithmic dependence, describes nonequilibrium fluctuations

Realization of the Landau definitions of effective Hamiltonian and nonequilibrium free energy in microscopic theory

Equilibrium fluctuations of some set of parameters in the states described by the canonical Gibbs distribution are investigated. In the theory of phase transitions of the second kind, these parameters are components of the order parameter. The microscopic realization of the Landau definition of the effective Hamiltonian of the system for studying the equilibrium fluctuations of the specified system of parameters is discussed in the terms of the probability density of their values. A general formula for this function is obtained and it is expressed through the equilibrium correlation functions of these parameters. An expression for the effective Hamiltonian in terms of deviations of the parameters from their equilibrium values is obtained. The deviations are considered small for conducting the calculations. The possibility of calculating the exact free energy of the system using the found effective Hamiltonian is discussed. In the microscopic theory, the implementation of the Landau definition of nonequilibrium thermodynamic potentials introduced in his phenomenological theory of phase transitions of the second kind is investigated. Nonequilibrium states of a fluctuating system described with some sets of parameters are considered. A general formula for nonequilibrium free energy expressed through the correlation functions of these parameters is obtained as for the effective Hamiltonian above. Like the previous case, the free energy expression via parameter deviations from the equilibrium values is obtained and small deviations are considered for calculations. The idea of the identity of the effective Hamiltonian of the system and its nonequilibrium free energy is discussed in connection with the Boltzmann distribution. The Gaussian approximation of both developed formalisms is considered. A generalization of the constructed theory for the case of spatially inhomogeneous states and the study of long-wave fluctuations are developed.

Thermal features of Barrow corrected-entropy black hole formulation

Through the last years, it was demonstrated that quantum corrections of entropy, represented by logarithmic and power law corrections terms, constituted an association between semi-classical entropic areas and the curvature correction in Einstein–Hilbert’s Lagrangian and vice-versa. Loop quantum gravity approach provided the logarithmic corrections, which arises from quantum and thermal equilibrium fluctuations. On the other hand, Barrow’s entropy was introduced from the fact that the black hole surface can be modified due to quantum gravitational effects. The new exponent $$\Delta $$Δ that appears in Barrow’s entropy is a measure of this perturbation. In this letter we have analyzed the thermodynamical effects of the quantum fluctuations upon the geometry of a Barrow’s black hole. We demonstrated that new formulations of the equipartition law, which corresponds to the horizon energy, can be constructed from both entropic formalisms. Besides, we have calculated the heat capacity for both formulations and we discussed their thermal viability. We have also establish a condition on one of the constant pre-factors of the logarithmic correction.