## Potential-Descending Principle as the First Principle of Statistical Physics

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

One main component of this paper is to postulate the following potential-descending principle (PDP) for statistical physics:

Potential-Descending Principle: For each thermodynamic system, there are order parameters ${u=(u_1, \cdots, u_N)}$, control parameters ${\lambda}$, and the thermodynamic potential functional ${F(u; \lambda)}$. For a non-equilibrium state ${u(t; u_0)}$ of the system with initial state ${u(0, u_0)=u_0}$, we have the following properties:

1)  the potential ${F(u(t; u_0); \lambda)}$ is decreasing:

$\displaystyle \frac{\text{d} }{\text{d} t} F(u(t; u_0); \lambda) < 0 \qquad \forall t > 0;$

2) the order parameters ${u(t; u_0)}$ have a limit

$\displaystyle \lim\limits_{t \rightarrow \infty}u(t; u_0) = \bar u;$

3)  there is an open and dense set ${\mathcal O}$ of initial data in the space of state functions, such that for any ${u_0 \in \mathcal O}$, the corresponding ${\bar u}$ is a minimum of ${F}$, which is called an equilibrium of the thermodynamic system:

$\displaystyle \delta F(\bar u;\lambda)= 0.$

1. In classical thermodynamics, the order parameters (state functions) ${u=(u_1, \cdots, u_N)}$, the control parameters ${\lambda}$, and the thermodynamic potential (or potential in short) ${F}$ are all treated as state variables. This way of mixing different level of physical quantities leads to difficulties for the understanding and the development of statistical physics.

One important feature of PDP above is the distinction of different levels of thermodynamical quantities— thermodynamical potentials are functionals of the order parameters (state functions), and orders parameters are functions of control parameters: potentials are first level physical quantities, order-parameters are on the second-level, and the control parameters are on the third level.

2. In classical thermodynamics, the first and second laws are treated as the first principles, from which one derives other statistical properties of thermodynamical systems. One perception is that potential-decreasing property can be derived from the first and second laws. However, in the derivations, there is a hidden assumption that at the equilibrium, there is a free-variable in each pair of (entropy ${S}$, temperature ${T}$) and (generalized force ${f}$, displacement ${X}$). Here the free-variables corresponds to order-parameters. We discovered that this assumption is mathematically equivalent to the potential-descending principle. As an example, we consider an internal energy of a thermodynamic system, classical theory asserts that the first and second laws are given by

$\displaystyle dU \le \frac{\partial U}{\partial S} dS + \frac{\partial U}{\partial X}dX, \ \ \ \ \ (1)$

where the equality represents the first laws, describing the equilibrium state, and inequality presents second law for non-equilibrium state. However, there is a hidden assumption in (1) that ${S}$ and ${X}$ are free variables, and

$\displaystyle \frac{\partial U}{\partial T} \le 0, \qquad \frac{\partial U}{\partial f}\le 0,$

where, again, the equality is for equilibrium state and the strict inequality is for non-equilibrium state. Then it is clear to see that this assumption is mathematically equivalent to PDP. In other words, the potential-decreasing property cannot be derived if we treat the first and second laws as the only fundamental principles of thermodynamics. Also, we demonstrate that the potential-descending principle leads to both the first and second laws of thermodynamics. Therefore we reach the following conclusion:

the potential-descending principle is a more fundamental principle then the first and second laws.

3. For a thermodynamical systems, PDP provides a dynamical law for the transformation of non-equilibrium states to equilibrium states: the dynamic equations of a thermodynamic system in non-equilibrium state take the form

$\displaystyle \frac{\text{d} u }{\text{d} t } = - A \delta F(u), \ \ \ \ \ (2)$

where ${A}$ is positive and symmetric coefficient matrix.

4. According to the entropy formula:

$\displaystyle S=k\ln W,$

and by the minimum potential principle as part of PDP:

$\displaystyle \delta F=0.$

Here

$\displaystyle F=U_0 -ST -\mu_1 N - \mu_2 E, \ \ \ \ \ (3)$

where ${U_0}$ is the internal energy, which is a constant, ${N}$ is the number of particles, ${\mu_1}$ and ${\mu_2}$ are Lagrangian multipliers, and ${E}$ is the total energy. For this system, the entropy ${S}$ is an order parameter, and temperature ${T}$ is a control parameter.

Then we can derive, with similar procedures as in Section 6.1 of [R. K. Pathria & Paul D. Beale, Statistical Mechanics, 3rd Edition, Elsevier, 2011], all three distributions: the Maxwell-Boltzmann distribution, the Fermi-Dirac distribution and the Bose-Einstein distribution. This shows that

the potential-descending principle is also the first principle of statistical mechanics.

Tian Ma & Shouhong Wang