arXiv subjects

Vaclav Klika

Publications and source records attributed to Vaclav Klika.

11 recordsLinked to original sources

arXiv2026

How the arrow of time emerges from Hamiltonian systems by our incomplete knowledge

How does the arrow of time (dissipative, irreversible behavior) emerge from time-reversible Hamiltonian mechanics? Two ingredients are needed: the underlying system must be ergodic or phase-mixing, and our knowledge of the system must be incomplete. When the detailed dynamics explores its phase space and stays close to a submanifold parametrized by a reduced set of state variables, the lack-of-fit reduction method reveals that the effective equations for those reduced variables are necessarily irreversible. To make this precise, we present a path-integral formulation of the lack-of-fit reduction in non-equilibrium thermodynamics, which shows how the GENERIC framework (reversible Hamiltonian part plus irreversible gradient flow) emerges from purely Hamiltonian mechanics without any fitting parameters. The formulation is based on the Onsager-Machlup variational principle, and it yields reduced dynamical equations by minimizing the information discrepancy between the detailed and reduced evolutions. Subsequently, the reduction method is illustrated on the Kac--Zwanzig model, confirming that dissipation emerges from ignoring degrees of freedom, and on diffusion, where diffusion equation for the hydrodynamic mass density emerges from Vlasov equation. We also show how to generalize the Fisher information matrix and Kullback--Leibler divergence to arbitrary concave entropies via the principle of maximum entropy, including non-Boltzmann-Gibbs cases such as the Tsallis--Havrda--Charvát entropy.

cond-mat.stat-mech↗

arXiv2024

Approaches to conservative Smoothed Particle Hydrodynamics with entropy

Smoothed particle hydrodynamics (SPH) is typically used for barotropic fluids, where the pressure depends only on the local mass density. Here, we show how to incorporate the entropy into the SPH, so that the pressure can also depend on the temperature, while keeping the growth of the total entropy, conservation of the total energy, and symplecticity of the reversible part of the SPH equations. The SPH system of ordinary differential equations with entropy is derived by means of the Poisson reduction and the Lagrange-Euler transformation. We present several approaches towards SPH with entropy, which are then illustrated on systems with discontinuities, on adiabatic and nonadiabatic expansion, and on the Rayleigh-Beenard convection without the Boussinesq approximation. Finally, we show how to model hyperbolic heat conduction within the SPH, extending the SPH variables with not only entropy but also a heat-flux-related vector field.

physics.flu-dyn↗

arXiv2022

Comparison of the Symmetric Hyperbolic Thermodynamically Compatible framework with Hamiltonian mechanics of binary mixtures

How to properly describe continuum thermodynamics of binary mixtures where each constituent has its own momentum? The Symmetric Hyperbolic Thermodynamically Consistent (SHTC) framework and Hamiltonian mechanics in the form of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) provide two answers, which are similar but not identical, and are compared in this article. They are compared both analytically and numerically on several levels of description, varying in the amount of detail. The GENERIC equations, stemming from the Liouville equation, contain terms expressing self-advection of the relative velocity by itself, which lead to a vorticity-dependent diffusion matrix after a reduction. The SHTC equations, on the other hand, do not contain such terms. We also show how to formulate a theory of mixtures with two momenta and only one temperature that is compatible with the Liouville equation and possesses the Hamiltonian structure, including Jacobi identity.

physics.flu-dyn↗

arXiv2021

Fluctuating Multiscale Mass Action Law

The classical mass action law in chemical kinetics is put into the context of multiscale thermodynamics.Despite the purely dissipative character of the classical mass action law, inertial effects also play a role in chemical kinetics. Therefore, the kinetics is extended to an enlarged state space with reaction rates as new state variables and bringing inertial effects. The dynamics is then lifted to the Liouville description within kinetic theory on the enlarged state space in order to include fluctuations. Subsequently, the kinetic theory is reduced to its first and second moments, leading to a new Grad-like hierarchy in chemical kinetics, expressing the multiscale nature of the chemical kinetic with inertia. Dissipation within the extended state space is proposed and it is shown to lead to the classical mass action law when the moments relax to their respective quasi-equilibria. In particular, we demonstrate the possibility of oscillating homogeneous chemical reactions and how the correlations of fluctuations correspond with the chemical kinetics.

physics.chem-ph↗

arXiv2020

Generalization of the dynamical lack-of-fit reduction

The lack-of-fit statistical reduction, developed and formulated first by Bruce Turkington, is a general method taking Liouville equation for probability density (detailed level) and transforming it to reduced dynamics of projected quantities (less detailed level). In this paper the method is generalized. The Hamiltonian Liouville equation is replaced by an arbitrary Hamiltonian evolution combined with gradient dynamics (GENERIC), the Boltzmann entropy is replaced by an arbitrary entropy, and the kinetic energy by an arbitrary energy. The gradient part is a generalized gradient dynamics generated by a dissipation potential. The reduced evolution of the projected state variables is shown to preserve the GENERIC structure of the original (detailed level) evolution. The dissipation potential is obtained by solving a Hamilton-Jacobi equation. In summary, the lack-of-fit reduction can start with GENERIC and obtain GENERIC for the reduced state variables.

cond-mat.stat-mech↗

arXiv2020

On Hamiltonian continuum mechanics

Continuum mechanics can be formulated in the Lagrangian frame (addressing motion of individual continuum particles) or in the Eulerian frame (addressing evolution of fields in an inertial frame). There is a canonical Hamiltonian structure in the Lagrangian frame. By transformation to the Eulerian frame we find the Poisson bracket for Eulerian continuum mechanics with deformation gradient (or the related distortion matrix). Both Lagrangian and Eulerian Hamiltonian structures are then discussed from the perspective of space-time variational formulation and by means of semidirect products and Lie algebras. Finally, we discuss the importance of the Jacobi identity in continuum mechanics and approaches to prove hyperbolicity of the evolution equations and their gauge invariance.

physics.class-ph↗

arXiv2019

Gradient and GENERIC evolution towards reduced dynamics

Let (M,J) be a dynamical model of macroscopic systems and (N,K) a less microscopic model (i.e. a model involving less details) of the same macroscopic systems; M and N are manifolds, J are vector fields on M, and K are vector fields on N. Let P be the phase portrait corresponding to (M,J) (i.e. P is the set of all trajectories in M generated by a family of vector fields in J), and R the phase portrait corresponding to (N,K). Thermodynamics in its general sense is a pattern recognition process in which R is recognized as a pattern in P. In particular, the classical (both equilibrium and nonequilibrium) thermodynamics arises in the investigation of relations between models (M,J) and models without time evolution, i.e. models with K= 0. In such case R is a submanifold of M composed of fixed points. Let Su mapping M to R be a potential, called an upper entropy, generating the vector field J. The equilibrium thermodynamic relation in N is the lower entropy Sd(y) defined by Sd(y)=Su(x=y), where x is in M,y in N, and y is given by a final destination (i.e. when the time goes to infinity) of x in the time evolution generated by the vector field J. In this paper we show that if K is not zero (e.g. in externally forced or, in other words, open systems), then the reduction also provides thermodynamics (we call it flux-thermodynamics). If certain conditions are satisfied, then the lower entropy Ss, that arises in the investigation of the approach of J to K, is the time derivative of the lower entropy Sd arising in the investigation of the approach of M to N as t goes to infinity.

math-ph↗

arXiv2019

Ehrenfest regularization of Hamiltonian systems

Imagine a freely rotating rigid body. The body has three principal axes of rotation. It follows from mathematical analysis of the evolution equations that pure rotations around the major and minor axes are stable while rotation around the middle axis is unstable. However, only rotation around the major axis (with highest moment of inertia) is stable in physical reality (as demonstrated by the unexpected change of rotation of the Explorer 1 probe). We propose a general method of Ehrenfest regularization of Hamiltonian equations by which the reversible Hamiltonian equations are equipped with irreversible terms constructed from the Hamiltonian dynamics itself. The method is demonstrated on harmonic oscillator, rigid body motion (solving the problem of stable minor axis rotation), ideal fluid mechanics and kinetic theory. In particular, the regularization can be seen as a birth of irreversibility and dissipation. In addition, we discuss and propose discretizations of the Ehrenfest regularized evolution equations such that key model characteristics (behavior of energy and entropy) are valid in the numerical scheme as well.

physics.comp-ph↗

arXiv2018

Entropy and Entropy Production in Multiscale Dynamics

Heat conduction is investigated on three levels: equilibrium, Fourier, and Cattaneo. The Fourier level is either the point of departure for investigating the approach to equilibrium or the final stage in the investigation of the approach from the Cattaneo level. Both investigations bring to the Fourier level an entropy and a thermodynamics. In the absence of external and internal influences preventing the approach to equilibrium the entropy that arises in the latter investigation is the production of the classical entropy that arises in the former investigation. If the approach to equilibrium is prevented, then the entropy that arises in the investigation of the approach from the Cattaneo level to the Fourier level still brings to the Fourier level the entropy and the thermodynamics even if the classical entropy and the classical thermodynamics is absent. We also note that vanishing total entropy production as a characterization of equilibrium state is insufficient.

cond-mat.stat-mech↗

arXiv2017

Statistical mechanics of Landau damping

Landau damping is the tendency of solutions to the Vlasov equation towards spatially homogeneous distribution functions. The distribution functions however approach the spatially homogeneous manifold only weakly, and Boltzmann entropy is not changed by Vlasov equation. On the other hand, density and kinetic energy density, which are integrals of the distribution function, approach spatially homogeneous states strongly, which is accompanied by growth of the hydrodynamic entropy. Such a behavior can be seen when Vlasov equation is reduced to the evolution equations for density and kinetic energy density by means of the Ehrenfest reduction.

physics.flu-dyn↗

arXiv2016

A hierarchy of Poisson brackets in non-equilibrium thermodynamics

Reversible evolution of macroscopic and mesoscopic systems can be conveniently constructed from two ingredients: an energy functional and a Poisson bracket. The goal of this paper is to elucidate how the Poisson brackets can be constructed and what additional features we also gain by the construction. In particular, the Poisson brackets governing reversible evolution in one-particle kinetic theory, kinetic theory of binary mixtures, binary fluid mixtures, classical irreversible thermodynamics and classical hydrodynamics are derived from Liouville equation. Although the construction is quite natural, a few examples where it does not work are included (e.g. the BBGKY hierarchy). Finally, a new infinite grand-canonical hierarchy of Poisson brackets is proposed, which leads to Poisson brackets expressing non-local phenomena such as turbulent motion or evolution of polymeric fluids. Eventually, Lie-Poisson structures standing behind some of the brackets are identified.

math-ph↗