Entropy

Entropy is an extensive state function satisfying

d S = δ Q_{rev} / T

for a reversible transfer of heat.

Table of contents

Specific entropy ( $s = S / M$ )

Specific entropy is entropy per unit mass. For a perfect gas with constant specific heat capacity

c_{v}

at constant volume and specific-heat ratio

γ

, differences obey

s_{2} - s_{1} = c_{v} lo g [\frac{p _{2}}{p _{1}} (\frac{ρ _{1}}{ρ _{2}})^{γ}],

where

p

is pressure and

ρ

is mass density. Thus a homentropic flow has constant

p / ρ^{γ}

.

Boltzmann entropy ( $S = k_{B} lo g Ω$ )

For a macrostate containing

Ω

equally likely microstates, the Boltzmann entropy is

S = k_{B} lo g Ω

.

Entropy density ( $s$ )

For a homogeneous equilibrium system with energy density

ρ

P

, chemical potential

μ

, and number density

n

, the entropy density is

s = \frac{ρ + P - μ n}{T} .

Entropy density at zero chemical potential ( $s = (ρ + P) / T$ )

For an extensive equilibrium system, the thermodynamic Euler relation gives

E = TS - P V + μ N

. At zero chemical potential, divide by

V

to obtain

s = (ρ + P) / T

. Equivalently,

d P = (ρ + P) d T / T

makes

d (\frac{( ρ + P ) V}{T}) = \frac{d ( ρ V ) + P d V}{T} .

The first law of thermodynamics identifies this as

d S

; the thermodynamic Euler relation fixes the entropy normalization. For adiabatic expansion satisfying the cosmological perfect-fluid continuity equation, the numerator on the right vanishes, proving cosmological entropy conservation.

Nonrelativistic entropy at zero chemical potential ( $s \sim g m^{3} (2 π)^{- 3/2} x^{- 1/2} e^{- x}$ )

For a dilute nonrelativistic gas with

x = m / T ≫ 1

and zero chemical potential, the Bose-Einstein distribution approaches the Maxwell-Boltzmann distribution. Expanding

E = m + p^{2} / (2 m) + \dots

in the number density integral gives

n \sim g (m T / (2 π))^{3/2} e^{- x}

. Since

ρ = mn + 3 n T /2 + \dots

and

P = n T

, the entropy density at zero chemical potential gives

s = n (x + 5/2 + \dots)

and hence the displayed leading expression. This includes the internal multiplicity

g

; a single real scalar field has

g = 1

.

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Wikipedia Bot  1

Entropy is a concept that appears in various fields, such as thermodynamics, information theory, and statistical mechanics. Its meaning can vary slightly depending on the context, but generally, it refers to a measure of disorder, uncertainty, or randomness in a system. 1. **Thermodynamics**: In thermodynamics, entropy is a measure of the amount of energy in a physical system that is not available to do work. It is often associated with the degree of disorder or randomness in a system.

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Ciro Santilli  40 Updated 2025-07-16

OK, can someone please just stop the philosophy and give numerical predictions of how entropy helps you predict the future?

The original notion of entropy, and the first one you should study, is the Clausius entropy.

For entropy in chemistry see: entropy of a chemical reaction.

The Unexpected Side of Entropy by Daan Frenkel

. Source. 2021.

The Biggest Ideas in the Universe | 20. Entropy and Information by Sean Carroll (2020)

Source. In usual Sean Carroll fashion, it glosses over the subject. This one might be worth watching. It mentions 4 possible definitions of entropy: Boltzmann, Gibbs, Shannon (information theory) and John von Neumann (quantum mechanics).

www.quantamagazine.org/what-is-entropy-a-measure-of-just-how-little-we-really-know-20241213/ What Is Entropy? A Measure of Just How Little We Really Know. on Quanta Magazine attempts to make the point that entropy is observer dependant. TODO details on that.

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