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:
to demonstrate that the PDP is the first principle to describe irreversibility of all thermodynamic systems;
to examine the problems faced by the Boltzmann equation.
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 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.
Third, the main source of this force field comes from the interaction between particles, and 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).
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.
This is a non-physical, rather arbitrary assumption, since 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
where () are constants, and . This shows that each steady-state is not stable, and this does not fit the reality.
Again, this is non-physical.
the Boltzmann is not a physical law, and also fails to achieve its two original goals.
Tian Ma & Shouhong Wang