Classical statistical mechanics

Entropy and the generalized second law

Statistical entropy S=kB ln Ω counts compatible microstates; the second law states that total entropy of an isolated system does not decrease.

S=k_B\ln\Omega, \qquad \Delta S_{\rm total}\ge 0

Definition: Quantities and meaning

Ω counts compatible microstates in a microcanonical model. For a general probability distribution pᵣ, Gibbs entropy is S=−kB Σᵣ pᵣ ln pᵣ. Statistical mechanics explains why equilibrium macroscopically dominates, while the second law is probabilistic for many-particle systems and does not forbid every small finite-time entropy fluctuation.

Entropy từ số vi trạng thái

Quantitative relation

S_{\rm Gibbs}=-k_B\sum_r p_r\ln p_r

Example: Worked example

An isolated system has Ω compatible states. If Ω increases from Ω₁ to Ω₂=4Ω₁, what is the entropy change?

Solution

ΔS=kB ln(Ω₂/Ω₁)=kB ln4.

Example: Example: entropy of mixing

Two equal portions of the same ideal gas, initially at the same temperature and pressure, are separated by a partition. Removing it causes no macroscopic change and ΔS=0 because identical particles are indistinguishable. For two different gases, free mixing gives ΔS_mix=−nR(x₁ln x₁+x₂ln x₂)>0. If x₁=x₂=1/2 and each gas has n moles, the increase is 2nR ln2. Distinguishing these cases avoids counting fictitious permutations of identical particles.

The second law also connects entropy with information about microstates: knowing the exact microstate leaves little uncertainty, while knowing only probabilities pᵣ gives a Gibbs entropy that quantifies missing information. For independent systems A and B, entropy is additive, S_AB=S_A+S_B; correlations require the joint entropy and invalidate naive addition. Microscopic dynamics of an isolated system can preserve information, yet macroscopic entropy tends to rise because coarse-grained descriptions discard fine correlations. Since equilibrium-compatible states dominate exponentially for large particle numbers, a substantial observed entropy decrease is extraordinarily unlikely, though not absolutely forbidden. A reversible process keeps total entropy constant; strict increase characterizes irreversible evolution. A subsystem may lose entropy if its surroundings gain a compensating amount. Always specify the system boundary.

Quick check

From S=kB ln Ω, what happens to entropy if Ω doubles?

What does the second law state for an isolated system?

References

Charles Kittel and Herbert Kroemer (1980). Thermal Physics
L. D. Landau and E. M. Lifshitz (1980). Statistical Physics