[Paper Review] Application of Bogolyubov's approach to the derivation of kinetic equations for dissipative systems

I. Goldhirsch, A. S. Peletminskii+2 more|arXiv (Cornell University)|Jul 12, 2013

Statistical Mechanics and Entropy3 citations

TL;DR

This paper extends Bogolyubov's functional hypothesis and correlation weakening principle to derive kinetic equations for classical dissipative many-body systems, particularly granular gases. By introducing a dissipation function within a Hamiltonian framework and applying the BBGKY hierarchy, the method reproduces the inelastic Boltzmann equation for dilute, monodisperse granular fluids with constant normal restitution, validating its applicability to dissipative systems through perturbative analysis.

ABSTRACT

The main goal of the present article is to extend the Bogolyubov method for deriving kinetic equations to dissipative many-body systems. The basic conjecture underlying the Bogolyubov approach is the functional hypothesis, according to which, the many-particle distribution functions are assumed to be functionals of the one-particle distribution function on kinetic time scales. Another ingredient in the Bogolyubov approach is the principle of the spatial weakening of correlations, which reflects statistical independence of physical values at distant spatial points. One can consider it as a reasonable mixing property of many-particle distribution functions. The motivation behind the generalization of Bogolyubov's approach to (classical) many-body dissipative systems is the wish to describe the dynamics of granular systems, in particular granular fluids. To this end we first define a general dissipative fluid through a dissipation function, thereby generalizing the commonly employed models for granular fluids. Using the Bogolyubov functional hypothesis we show how a reduction of the pertinent BBGKY hierarchy can be achieved. The method is then employed to cases which can be treated perturbatively, such as those in which the interactions are weak or the dissipation is small or the particle density is small. Kinetic descriptions are obtained in all of these limiting cases. As a test case, we show that the Bogolyubov method begets the now standard inelastic Boltzmann equation for dilute monodisperse collections of spheres whose collisions are characterized by a fixed coefficient of normal restitution. Possible further applications and implications are discussed.

Motivation & Objective

• To generalize Bogolyubov's approach—based on the functional hypothesis and spatial correlation weakening—to classical dissipative many-body systems, especially granular fluids.
• To develop a BBGKY hierarchy for dissipative systems using a dissipation function within a Hamiltonian formulation, overcoming limitations of pseudo-Liouville equations.
• To demonstrate the method's validity by recovering the standard inelastic Boltzmann equation for dilute, monodisperse granular gases with constant normal restitution.
• To explore the potential of this approach for studying dense granular systems, binary mixtures, and regimes with strong gradients where conventional methods fail.

Proposed method

• Introduce a dissipation function to generalize Hamiltonian mechanics for classical dissipative systems, enabling a consistent derivation of the BBGKY hierarchy.
• Apply Bogolyubov's functional hypothesis, assuming many-particle distribution functions are functionals of the one-particle distribution on kinetic time scales.
• Invoke the principle of spatial correlation weakening, modeling statistical independence at distant spatial points as a mixing property.
• Use perturbative expansions for weak interactions, small dissipation, or low density to reduce the BBGKY hierarchy to a closed kinetic equation.
• Derive the collision integral from asymptotic momenta and coordinates in the two-particle dynamics, including the Jacobian transformation under inelastic collisions.
• Verify consistency by showing that the derived collision integral matches the standard inelastic Boltzmann equation under the condition of fixed normal restitution.

Experimental results

Research questions

• RQ1 Can Bogolyubov's method be extended to derive kinetic equations for dissipative many-body systems, particularly granular gases?
• RQ2 How can a dissipation function be incorporated into a Hamiltonian formulation to describe inelastic collisions in classical systems?
• RQ3 Does the functional hypothesis and correlation weakening principle remain valid for dissipative systems, and can they yield consistent kinetic equations?
• RQ4 Can the derived kinetic equation reproduce the known inelastic Boltzmann equation for dilute granular gases with constant normal restitution?
• RQ5 What are the implications of this approach for studying dense granular systems or binary mixtures where standard kinetic theories fail?

Key findings

• The method successfully reproduces the inelastic Boltzmann equation for dilute, monodisperse granular gases with a fixed coefficient of normal restitution.
• The collision integral derived from the asymptotic dynamics, including the Jacobian transformation under inelastic collisions, exactly matches the standard form of the inelastic Boltzmann collision integral.
• The functional hypothesis and spatial correlation weakening principle are validated as viable assumptions for dissipative systems, enabling reduction of the BBGKY hierarchy.
• The approach is applicable to perturbative regimes such as weak interactions, small dissipation, or low particle density, yielding consistent kinetic descriptions.
• The formulation provides a new framework for studying granular systems with strong gradients or many-body contacts, where traditional methods face limitations.
• The method offers a potential alternative to the Enskog-Boltzmann equation for dense or complex granular mixtures, particularly where naive applications fail.