## Statistical Theory of Heat

Tian Ma & Shouhong Wang, Statistical Theory of Heat, Indiana University ISCAM Preprint #1711, 37 pages, August 26, 2017

The above paper presents a new statistical theory of heat based on the statistical theory of thermodynamics and on the recent developments of quantum physics.

### Main Motivations

First, by classical thermodynamics, for a thermodynamic system, the thermal energy ${Q_0}$ is given by

$\displaystyle Q_0=ST. \ \ \ \ \ (1)$

Here ${T}$ is the temperature of the system and ${S}$ is the entropy given by the Boltzmann formula:

$\displaystyle S=k\ln W, \ \ \ \ \ (2)$

where ${k}$ is the Boltzmann constant, and ${W}$ is the number of microscopic configurations of the system. It is then clear that in modern thermodynamics, there is simply no physical heat carrier in both the temperature ${T}$ and the entropy ${S}$, and hence there is no physical carrier for thermal energy ${Q_0=ST}$. Due to the lack of physical carrier in temperature ${T}$ and ${S=k ln W}$, and consequently in the thermal energy ${Q=ST}$, the nature of heat is still not fully understood.

The second motivation is the recent development on the photon cloud structure of electrons: the naked electron are surrounded by a shell layer cloud of photons. Therefore, electrons and photons form a natural conjugate pair of physical carriers for emission and absorption, reminiscent to the conjugation between ${T}$ and ${S}$.

Third, in view of ${Q=ST}$ and the above conjugation between photons and electrons, a theory of heat has to make

connections between 1) conjugation between electrons and photons, and 2) conjugation between temperature and entropy.

The new theory in this paper provides precisely such a connection: at the equilibrium of absorption and radiation, the average energy level of the system maintains unchanged, and represents the temperature of the system; at the same time, the number (density) of photons in the sea of photons represents the entropy (entropy density) of the system.

## Main Results

### 1. Energy level formula of temperature

In view of the above connection between the two conjugations, temperature must be associated with the energy levels of electrons, since it is an intensive physical quantity measuring certain strength of heat, reminiscent of the basic characteristic of energy levels of electrons. Also notice that there are abundant orbiting electrons and energy levels of electrons in atoms and molecules. Hence the energy levels of orbiting electrons, together with the kinetic energy of the system particles, provide a truthful representation of the system particles.

We derive the following energy level formula of temperature using the well-known Maxwell-Boltzmann, the Fermi-Dirac, and the Bose-Einstein distributions:

• for classical systems,

$\displaystyle kT= \sum_n\bigg(1-\frac{a_n}{N}\bigg)\frac{a_n\varepsilon_n} {N(1+\beta_n\ln\varepsilon_n)}, \ \ \ \ \ (3)$

• for system of bose particles,

$\displaystyle kT= \sum_n \bigg(1 + \frac{a_n}{g_n}\bigg) \frac{a_n\varepsilon_n}{N(1+\beta_n\ln\varepsilon_n)}; \ \ \ \ \ (4)$

• for systems of fermi particls,

$\displaystyle kT= \sum_n \bigg(1 - \frac{a_n}{g_n}\bigg) \frac{a_n\varepsilon_n}{N(1+\beta_n\ln\varepsilon_n)}. \ \ \ \ \ (5)$

Here ${\varepsilon_n}$ are the energy levels of the system particles, ${N}$ is the total number of particles, ${g_n}$ are the degeneracy factors (allowed quantum states) of the energy level ${\varepsilon_n}$, and ${a_n}$ are the distributions, representing the number of particles on the energy level ${\varepsilon_n}$.

The above formulas amount to saying that temperature is simply the (weighted) average energy level of the system. Also these formulas enable us to have a better understanding on the nature of temperature.

In summary, the nature of temperature ${T}$ is the (weighted) average energy level. Also, the temperature ${T}$ is a function of distributions ${\{a_n\}}$ and the energy levels ${\{\varepsilon_n\}}$ with the parameters ${\{\beta_n\}}$ reflecting the property of the material.

### 2. Photon number formula of entropy

In view of the conjugate relations, since entropy ${S}$ is an extensive variable, we need to characterize entropy as the number of photons in the photon gas between system particles, or the photon density of the photon gas in the system. Also, photons are Bosons and obey the Bose-Einstein distribution. Then we can make a connection between entropy and the number of photons and derive

$\displaystyle S = kN_0 \left[ 1+ \frac{1}{kT} \sum_n \frac{\varepsilon_n a_n}{N_0}\right], \ \ \ \ \ (6)$

where ${\varepsilon_n}$ are the energy levels of photons, and ${a_n}$ are the distribution of photons at energy level ${\varepsilon_n}$, ${N_0=\sum_n a_n}$ is the total number of photons between particles in the system, and ${\sum_n \frac{\varepsilon_n}{kT}a_n}$ represents the number of photons in the sense of average energy level.

It is worth mentioning that this new entropy formula is equivalent to the Boltzmann entropy formula ${S=k\ln W}$. However, their physical meanings have changed: the new formula (6) provides explicitly that

the physical carrier of heat is the photons.

### 3. Temperature theorem

By the temperature and the entropy formulas (5) and (6), we arrive immediately at the following results of temperature:

• There are minimum and maximum values of temperature with ${T_{\min}=0}$ and ${T_{\max}}$;
• When the number of photons in the system is zero, the temperature is at absolute zero; namely, the absence of photons in the system is the physical reason causing absolute zero temperature;
• (Nernst Theorem) With temperature at absolute zero, the entropy of the system is zero;
• With temperature at absolute zero, all particles fills all lowest energy levels.

### 4. Thermal energy formula

Thanks to the entropy formula (6), we derive immediately the following thermal energy formula:

$\displaystyle Q_0 = ST = E_0 + k N_0 T, \ \ \ \ \ (7)$

where ${E_0=\sum_n a_n \varepsilon_n}$ is the total energy of photons in the system, ${\varepsilon_n}$ are the energy levels of photons, and ${a_n}$ are the distribution of photons at energy level ${\varepsilon_n}$, and ${N_0}$ is the number of photons in the system.

The theory of heat presented in this paper is established based on physical theories on fundamental interactions, the photon cloud model of electrons, the first law of thermodynamics, statistical theory of thermodynamics, radiation mechanism of photons, and energy level theory of micro-particles. The theory utilizes rigorous mathematics to reveal the physical essence of temperature, entropy and heat.

Tian Ma & Shouhong Wang