## Irreversibility and Problems in the Boltzmann Equation

Tian Ma & Shouhong Wang, Dynamical Law of Physical Motion and Potential-Descending Principle, The Institute for Scientific Computing and Applied Mathematics Preprint #1701, July 6, 2017

The previous blog is on postulating the potential-descending principle (PDP) as the first principle of statistical physics. The purpose of this blog is on the next two components of the above paper:

1. to demonstrate that the PDP is the first principle to describe irreversibility of all thermodynamic systems;

2. to examine the problems faced by the Boltzmann equation.

Irreversibility

First, irreversibility is a macroscopic property of thermodynamics, and must be described by the first level physical quantities–thermodynamic potentials, rather than the second level quantities (state functions) or the third level quantities (control parameters).

Second, entropy ${S}$ is a state function, which is the solution of basic thermodynamic equations. Thermodynamic potential is a higher level physical quantity than entropy, and consequently, is the correct physical quantity for describing irreversibility for all thermodynamic systems.

Problems in Boltzmann Equation

Historically, great effort has been put on establishing a mathematical model of entropy-increasing principle. The Boltzmann equation is introduced mainly for this purpose. Since there is no first principle to achieve this purpose, the Boltzmann equation is introduced as a phenomenological model with two specific goals:

• to derive the entropy-ascending principle, and

• to make the Maxwell-Boltzmann distribution (realistic equilibrium state of a dilute gaseous system) a steady-state solution.

However, the Boltzmann equation faces many problems:

First,  laws of physics (equations) should not use state functions, which are themselves governed by physical laws, as independent variables. The Boltzmann equation violates this simple physical rule by using the velocity field as an independent variable. Consequently, the Boltzmann equation is not a physical law.

Second, the Boltzmann equation uses the velocity field ${v}$ as an independent variable. This leads to a new unknown function (force field) in the Boltzmann equation:

$\displaystyle F=F(t, x, v), \ \ \ \ \ (1)$

which is the sum of the external force and the force generated by the total interaction potential of all particles in the system, including the force due to collision.

Third, the main source of this force field ${F}$ comes from the interaction between particles, and ${F}$ is realistically not zero. Otherwise, all particles in the system would make uniform rectilinear motion, and in particular there would be no particle collisions (collision is close-distance interactions).

Fourth, if

$\displaystyle F\not=0, \ \ \ \ \ (2)$

then the Maxwell distribution fails to be the steady state solution of the Boltzmann equation. Since the Maxwell-distribution is the realistic equilibrium state of dilute gaseous systems, the Boltzmann equation fails to describe the underlying physical phenomena in this important regard.

Fifth, in deriving the H-Theorem (i.e. the entropy-ascending principle), the following must be assumed:

$\displaystyle F=F(t, x) \text{ \it is independent of } v. \ \ \ \ \ (3)$

This is a non-physical, rather arbitrary assumption, since ${F}$ includes the force generated by all interactions (including collision) and must be velocity dependent. Consequently the H-Theorem is not a natural consequence of the Boltzmann equation.

Sixth, ignoring the non-physical nature of assumption (2), the space of all steady state solutions of the Boltzmann equation is of five-dimensional. Namely, the general form of the steady state solutions is

$\displaystyle \bar \rho = e^{\alpha_0 + \alpha_1 v_1 + \alpha_2 v_2 + \alpha_3 v_3 + \alpha_4 v^2},$

where ${\alpha_i}$ (${i=0, \cdots, 4}$) are constants, and ${v=(v_1, v_2, v_3)}$. This shows that each steady-state is not stable, and this does not fit the reality.

Seventh, the entropy-increasing principle shows that a gaseous system in the equilibrium has the maximum entropy, i.e. the Maxwell distribution should be the maximum of

$\displaystyle S=-\int \rho\ln \rho dx dv + S_0, \ \ \ \ \ (4)$

However, the maximum of ${S}$ is given by

$\displaystyle \rho_0=e^{-1}.$

Again, this is non-physical.

In summary,

the Boltzmann is not a physical law, and also fails to achieve its two original goals.

Tian Ma & Shouhong Wang