A
thermodynamic
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
free entropy is an entropic
thermodynamic potential analogous to the
free energy. Also known as a Massieu, Planck, or Massieu–Planck potentials (or functions), or (rarely) free information. In
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, free entropies frequently appear as the logarithm of a
partition function. The
Onsager reciprocal relations
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.
"Reciprocal relations" occur betwe ...
in particular, are developed in terms of entropic potentials. In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, free entropy means something quite different: it is a generalization of entropy defined in the subject of
free probability Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products.
This theory was in ...
.
A free entropy is generated by a
Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
of the entropy. The different potentials correspond to different constraints to which the system may be subjected.
Examples
The most common examples are:
where
::
is
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
::
is the Massieu potential
[
]
::
is the Planck potential
::
is
internal energy
::
is
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
::
is
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
::
is
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
::
is
Helmholtz free energy
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz ener ...
::
is
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and ...
::
is
number of particles
The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter ''N'', is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is ...
(or number of moles) composing the ''i''-th chemical component
::
is the
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
of the ''i''-th
chemical component
::
is the total number of
components
::
is the
th components.
Note that the use of the terms "Massieu" and "Planck" for explicit Massieu-Planck potentials are somewhat obscure and ambiguous. In particular "Planck potential" has alternative meanings. The most standard notation for an entropic potential is
, used by both
Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical p ...
and
Schrödinger. (Note that Gibbs used
to denote the free energy.) Free entropies where invented by French engineer
François Massieu François Jacques Dominique Massieu (4 August 1832 – 5 February 1896) was a French thermodynamics engineer noted for his two 1869 characteristic functions, each of which known as a Massieu function (the first of which sometimes called free entrop ...
in 1869, and actually predate Gibbs's free energy (1875).
Dependence of the potentials on the natural variables
Entropy
:
By the definition of a total differential,
:
From the
equations of state,
:
The differentials in the above equation are all of
extensive variables, so they may be integrated to yield
:
Massieu potential / Helmholtz free entropy
:
:
:
Starting over at the definition of
and taking the total differential, we have via a Legendre transform (and the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
)
:
:
:
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From
we see that
:
If reciprocal variables are not desired,
[
]
:
:
:
:
:
Planck potential / Gibbs free entropy
:
:
:
Starting over at the definition of
and taking the total differential, we have via a Legendre transform (and the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
)
:
:
:
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From
we see that
:
If reciprocal variables are not desired,
:
:
:
:
:
References
Bibliography
*
*{{cite book
, first = Herbert B. , last = Callen , author-link = Herbert Callen , year = 1985
, title = Thermodynamics and an Introduction to Thermostatistics , edition = 2nd
, publisher = John Wiley & Sons , location = New York , isbn = 0-471-86256-8
Thermodynamic entropy