May 8, 2024 by gary_mail@hotmail.com

Statistical entropy

Statistical entropy is a concept in thermodynamics that quantifies the number of different ways a system can be arranged while still maintaining its macroscopic properties.

The total energy of a system, $E$ , is given by eq5. Its total differential is:

$dE=\sum_i\varepsilon_idn_i+\sum_in_id\varepsilon_i$

Consider the transfer of heat to a closed system of particles. If the volume of the system is constant, then no work is done on the system and the energy level of each state is unchanged $d\varepsilon_i=0$ . However, the population of particles in each state $dn_i$ varies according to temperature. Hence,

$dE=\sum_i\varepsilon_idn_i\;\;\;\;\;\;\;\;26$

If we interpret $E$ as the internal energy of the system, $U$ , then according to the first law of thermodynamics of a constant volume system,

$dE=dU=dq\;\;\;\;\;\;\;\;27$

For a reversible transfer of heat, the second law of thermodynamics states that ${dS=\frac{dq}{T}}$ . Combining $dS$ , eq26 and eq27,

$dS=k\beta\sum_i\varepsilon_idn_i\;\;\;\;\;\;\;\;28$

Substituting eq16 in eq28,

$dS=k\sum_i\biggr\(\frac{\partial lnW}{\partial n_i}\biggr\)dn_i+k\alpha\sum_idn_i$

Since the total number of particles for a closed system is constant, ${\sum_idn_i=0}$ and

$dS=k\sum_i\biggr\(\frac{\partial lnW}{\partial n_i}\biggr\)dn_i\;\;\;\;\;\;\;\;29$

Substituting eq4 in eq29, we have $dS=kdlnW$ or the integrated form:

$S=klnW\;\;\;\;\;\;\;\;30$

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