In

_{''j''} of particles of each type ''j''. The differential of the free energy then generalizes to
:$dF\; =\; -S\backslash ,dT\; -\; P\backslash ,dV\; +\; \backslash sum\_j\; \backslash mu\_j\backslash ,dN\_j,$
where the $N\_j$ are the numbers of particles of type ''j'', and the $\backslash mu\_j$ are the corresponding

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Helmholtz and Gibbs Free Energies

{{Authority control Physical quantities Hermann von Helmholtz State functions Thermodynamic free energy

thermodynamics
Thermodynamics is a branch of physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ot ...

, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential
A thermodynamic potential (or more accurately, a thermodynamic potential energy)ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.4 Helmholtz energy, Helmholtz functionISO/IEC 80000-5, Quantities an units, Part 5 - Thermodyn ...

that measures the useful work
Work may refer to:
* Work (human activity)
Work or labor is intentional activity people perform to support themselves, others, or the needs and wants of a wider community. Alternatively, work can be viewed as the human activity that cont ...

obtainable from a closed thermodynamic system
A thermodynamic system is a body of matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...

at a constant temperature
Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy
Thermal radiation in visible light can be seen on this hot metalwork.
Thermal energy refers to several distinct physical concept ...

(isothermal
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qua ...

). The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which temperature is held constant. At constant temperature, the Helmholtz free energy is minimized at equilibrium.
In contrast, the Gibbs free energy
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qua ...

or free enthalpy is most commonly used as a measure of thermodynamic potential (especially in chemistry
Chemistry is the scientific
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
A fact is an occurrence in the real world. T ...

) when it is convenient for applications that occur at constant ''pressure''. For example, in explosives
An explosive (or explosive material) is a reactive substance that contains a great amount of potential energy that can produce an explosion
An explosion is a rapid expansion in volume
Volume is a expressing the of enclosed by a . ...

research Helmholtz free energy is often used, since explosive reactions by their nature induce pressure changes. It is also frequently used to define fundamental equations of state
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...

of pure substances.
The concept of free energy was developed by Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist
A physicist is a scientist
A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branc ...

, a German physicist, and first presented in 1882 in a lecture called "On the thermodynamics of chemical processes". From the German word ''Arbeit'' (work), the International Union of Pure and Applied Chemistry
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations that represents chemists in individual countries. It is a member of the International Science Council (ISC). IUPAC ...

(IUPAC) recommends the symbol ''A'' and the name ''Helmholtz energy''. In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, the symbol ''F'' is also used in reference to ''free energy'' or ''Helmholtz function''.
Definition

The Helmholtz energy is defined as $$F\; \backslash equiv\; U\; -\; TS,$$ where * ''F'' is the Helmholtz free energy (sometimes also called ''A'', particularly in the field ofchemistry
Chemistry is the scientific
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
A fact is an occurrence in the real world. T ...

) (: joule
The joule ( ; symbol: J) is a derived unit of energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates ...

s, CGS: erg
The erg is a unit of energy equal to 10−7joule
The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an objec ...

s),
* ''U'' is the internal energy
The internal energy of a thermodynamic system
A thermodynamic system is a body of matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that ca ...

of the system (SI: joules, CGS: ergs),
* ''T'' is the absolute temperature (kelvin
The kelvin is the base unit of temperature
Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy
Thermal radiation in visible light can be seen on this hot metalwork.
Thermal en ...

s) of the surroundings, modelled as a heat bath,
* ''S'' is the 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 thermodynamic ...

of the system (SI: joules per kelvin, CGS: ergs per kelvin).
The Helmholtz energy is the Legendre transformation
In mathematics and physics, the Legendre transformation, named after Adrien-Marie Legendre, is an involution (mathematics), involutive List of transforms, transformation on the real number, real-valued convex functions of one real variable. In phys ...

of the internal energy ''U'', in which temperature replaces entropy as the independent variable.
Formal development

Thefirst law of thermodynamics
The first law of thermodynamics is a version of the law of conservation of energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department o ...

in a closed system provides
:$\backslash mathrmU\; =\; \backslash delta\; Q\backslash \; +\; \backslash delta\; W,$
where $U$ is the internal energy, $\backslash delta\; Q$ is the energy added as heat, and $\backslash delta\; W$ is the work done on the system. The second law of thermodynamics
The second law of thermodynamics establishes the concept of 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 an ...

for a reversible process yields $\backslash delta\; Q\; =\; T\backslash ,\backslash mathrmS$. In case of a reversible change, the work done can be expressed as $\backslash delta\; W\; =\; -p\backslash ,\backslash mathrmV$ (ignoring electrical and other non-''PV'' work):
:$\backslash mathrmU\; =\; T\backslash ,\backslash mathrmS\; -\; p\backslash ,\backslash mathrmV.$
Applying the product rule for differentiation to d(''TS'') = ''T'' d''S'' + ''S'' d''T'', it follows
:$\backslash mathrmU\; =\; \backslash mathrm(TS)\; -\; S\backslash ,\backslash mathrmT\; -\; p\backslash ,\backslash mathrmV,$
and
:$\backslash mathrm(U\; -\; TS)\; =\; -S\backslash ,\backslash mathrmT\; -\; p\backslash ,\backslash mathrmV.$
The definition of ''F'' = ''U'' − ''TS'' enables to rewrite this as
:$\backslash mathrmF\; =\; -S\backslash ,\backslash mathrmT\; -\; p\backslash ,\backslash mathrmV.$
Because ''F'' is a thermodynamic function of state
In the Thermodynamics#Equilibrium_thermodynamics, thermodynamics of equilibrium, a state function, function of state, or point function is a function defined for a system relating several state variables or state quantities that depends only on the ...

, this relation is also valid for a process (without electrical work or composition change) that is not reversible, as long as the system pressure and temperature are uniform.
Minimum free energy and maximum work principles

The laws of thermodynamics are most easily applicable to systems undergoing reversible processes or processes that begin and end in thermal equilibrium, although irreversible quasistatic processes or spontaneous processes in systems with uniform temperature and pressure (u''PT'' processes) can also be analyzed based on thefundamental thermodynamic relation
In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentiall ...

as shown further below. First, if we wish to describe phenomena like chemical reactions, it may be convenient to consider suitably chosen initial and final states in which the system is in (metastable) thermal equilibrium. If the system is kept at fixed volume and is in contact with a heat bath at some constant temperature, then we can reason as follows.
Since the thermodynamical variables of the system are well defined in the initial state and the final state, the internal energy increase $\backslash Delta\; U$, the 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 thermodynamic ...

increase $\backslash Delta\; S$, and the total amount of work that can be extracted, performed by the system, $W$, are well defined quantities. Conservation of energy implies
:$\backslash Delta\; U\_\backslash text\; +\; \backslash Delta\; U\; +\; W\; =\; 0.$
The volume of the system is kept constant. This means that the volume of the heat bath does not change either, and we can conclude that the heat bath does not perform any work. This implies that the amount of heat that flows into the heat bath is given by
:$Q\_\backslash text\; =\; \backslash Delta\; U\_\backslash text\; =\; -(\backslash Delta\; U\; +\; W).$
The heat bath remains in thermal equilibrium at temperature ''T'' no matter what the system does. Therefore, the entropy change of the heat bath is
:$\backslash Delta\; S\_\backslash text\; =\; \backslash frac\; =\; -\backslash frac.$
The total entropy change is thus given by
:$\backslash Delta\; S\_\backslash text\; +\; \backslash Delta\; S\; =\; -\backslash frac.$
Since the system is in thermal equilibrium with the heat bath in the initial and the final states, ''T'' is also the temperature of the system in these states. The fact that the system's temperature does not change allows us to express the numerator as the free energy change of the system:
:$\backslash Delta\; S\_\backslash text\; +\; \backslash Delta\; S\; =\; -\backslash frac.$
Since the total change in entropy must always be larger or equal to zero, we obtain the inequality
:$W\; \backslash leq\; -\backslash Delta\; F.$
We see that the total amount of work that can be extracted in an isothermal process is limited by the free-energy decrease, and that increasing the free energy in a reversible process requires work to be done on the system. If no work is extracted from the system, then
:$\backslash Delta\; F\; \backslash leq\; 0,$
and thus for a system kept at constant temperature and volume and not capable of performing electrical or other non-''PV'' work, the total free energy during a spontaneous change can only decrease.
This result seems to contradict the equation d''F'' = −''S'' d''T'' − ''P'' d''V'', as keeping ''T'' and ''V'' constant seems to imply d''F'' = 0, and hence ''F'' = constant. In reality there is no contradiction: In a simple one-component system, to which the validity of the equation d''F'' = −''S'' d''T'' − ''P'' d''V'' is restricted, no process can occur at constant ''T'' and ''V'', since there is a unique ''P''(''T'', ''V'') relation, and thus ''T'', ''V'', and ''P'' are all fixed. To allow for spontaneous processes at constant ''T'' and ''V'', one needs to enlarge the thermodynamical state space of the system. In case of a chemical reaction, one must allow for changes in the numbers ''N''chemical potential
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is governed ...

s. This equation is then again valid for both reversible and non-reversible u''PT'' changes. In case of a spontaneous change at constant ''T'' and ''V'' without electrical work, the last term will thus be negative.
In case there are other external parameters, the above relation further generalizes to
:$dF\; =\; -S\backslash ,dT\; -\; \backslash sum\_i\; X\_i\backslash ,dx\_i\; +\; \backslash sum\_j\; \backslash mu\_j\backslash ,dN\_j.$
Here the $x\_i$ are the external variables, and the $X\_i$ the corresponding generalized forcesGeneralized forces find use in Lagrangian mechanics
Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia

.
Relation to the canonical partition function

A system kept at constant volume, temperature, and particle number is described by thecanonical ensemble
In statistical mechanics
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, i ...

. The probability of finding the system in some energy eigenstate ''r'', for any microstate ''i'', is given by
$$P\_r\; =\; \backslash frac,$$
where
*$\backslash beta\; =\; \backslash frac,$
*$E\_r$ is the energy of accessible state $r$
*$Z\; =\; \backslash sum\_i\; e^.$
''Z'' is called the partition function of the system. The fact that the system does not have a unique energy means that the various thermodynamical quantities must be defined as expectation values. In the thermodynamical limit of infinite system size, the relative fluctuations in these averages will go to zero.
The average internal energy of the system is the expectation value of the energy and can be expressed in terms of ''Z'' as follows:
:$U\; \backslash equiv\; \backslash langle\; E\; \backslash rangle\; =\; \backslash sum\_r\; P\_r\; E\_r\; =\; \backslash sum\_r\; \backslash frac\; =\; \backslash sum\_r\; \backslash frac\; =\; \backslash frac\; =\; -\backslash frac.$
If the system is in state ''r'', then the generalized force corresponding to an external variable ''x'' is given by
:$X\_r\; =\; -\backslash frac.$
The thermal average of this can be written as
:$X\; =\; \backslash sum\_r\; P\_r\; X\_r\; =\; \backslash frac\; \backslash frac.$
Suppose that the system has one external variable $x$. Then changing the system's temperature parameter by $d\backslash beta$ and the external variable by $dx$ will lead to a change in $\backslash log\; Z$:
:$d(\backslash log\; Z)\; =\; \backslash frac\backslash ,d\backslash beta\; +\; \backslash frac\backslash ,dx\; =\; -U\backslash ,d\backslash beta\; +\; \backslash beta\; X\backslash ,dx.$
If we write $U\backslash ,d\backslash beta$ as
:$U\backslash ,d\backslash beta\; =\; d(\backslash beta\; U)\; -\; \backslash beta\backslash ,\; dU,$
we get
:$d(\backslash log\; Z)\; =\; -d(\backslash beta\; U)\; +\; \backslash beta\backslash ,\; dU\; +\; \backslash beta\; X\; \backslash ,dx.$
This means that the change in the internal energy is given by
:$dU\; =\; \backslash frac\backslash ,d(\backslash log\; Z\; +\; \backslash beta\; U)\; -\; X\backslash ,dx.$
In the thermodynamic limit, the fundamental thermodynamic relation
In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentiall ...

should hold:
:$dU\; =\; T\backslash ,\; dS\; -\; X\backslash ,\; dx.$
This then implies that the entropy of the system is given by
:$S\; =\; k\backslash log\; Z\; +\; \backslash frac\; +\; c,$
where ''c'' is some constant. The value of ''c'' can be determined by considering the limit ''T'' → 0. In this limit the entropy becomes $S\; =\; k\; \backslash log\; \backslash Omega\_0$, where $\backslash Omega\_0$ is the ground-state degeneracy. The partition function in this limit is $\backslash Omega\_0\; e^$, where $U\_0$ is the ground-state energy. Thus, we see that $c\; =\; 0$ and that
:$F\; =\; -kT\backslash log\; Z.$
Relating free energy to other variables

Combining the definition of Helmholtz free energy :$F\; =\; U\; -\; T\; S$ along with the fundamental thermodynamic relation $$dF\; =\; -S\backslash ,dT\; -\; P\backslash ,dV\; +\; \backslash mu\backslash ,dN,$$ one can find expressions for entropy, pressure and chemical potential: :$S\; =\; \backslash left.-\backslash left(\; \backslash frac\; \backslash right)\; \backslash \_,\; \backslash quad\; P\; =\; \backslash left.-\backslash left(\; \backslash frac\; \backslash right)\; \backslash \_,\; \backslash quad\; \backslash mu\; =\; \backslash left.\backslash left(\; \backslash frac\; \backslash right)\; \backslash \_.$ These three equations, along with the free energy in terms of the partition function, :$F\; =\; -kT\backslash log\; Z,$ allow an efficient way of calculating thermodynamic variables of interest given the partition function and are often used in density of state calculations. One can also doLegendre transformation
In mathematics and physics, the Legendre transformation, named after Adrien-Marie Legendre, is an involution (mathematics), involutive List of transforms, transformation on the real number, real-valued convex functions of one real variable. In phys ...

s for different systems. For example, for a system with a magnetic field or potential, it is true that
:$m\; =\; \backslash left.-\backslash left(\; \backslash frac\; \backslash right)\; \backslash \_,\; \backslash quad\; V\; =\; \backslash left.\backslash left\; (\; \backslash frac\; \backslash right)\; \backslash \_.$
Bogoliubov inequality

Computing the free energy is an intractable problem for all but the simplest models in statistical physics. A powerful approximation method ismean-field theory
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

, which is a variational method based on the Bogoliubov inequality. This inequality can be formulated as follows.
Suppose we replace the real Hamiltonian $H$ of the model by a trial Hamiltonian $\backslash tilde$, which has different interactions and may depend on extra parameters that are not present in the original model. If we choose this trial Hamiltonian such that
:$\backslash left\backslash langle\backslash tilde\backslash right\backslash rangle\; =\; \backslash langle\; H\; \backslash rangle,$
where both averages are taken with respect to the canonical distribution defined by the trial Hamiltonian $\backslash tilde$, then
:$F\; \backslash leq\; \backslash tilde,$
where $F$ is the free energy of the original Hamiltonian, and $\backslash tilde$ is the free energy of the trial Hamiltonian. By including a large number of parameters in the trial Hamiltonian and minimizing the free energy, we can expect to get a close approximation to the exact free energy.
The Bogoliubov inequality is often formulated in a slightly different but equivalent way. If we write the Hamiltonian as
:$H\; =\; H\_0\; +\; \backslash Delta\; H,$
where $H\_0$ is exactly solvable, then we can apply the above inequality by defining
:$\backslash tilde\; =\; H\_0\; +\; \backslash langle\backslash Delta\; H\backslash rangle\_0.$
Here we have defined $\backslash langle\; X\backslash rangle\_0$ to be the average of ''X'' over the canonical ensemble defined by $H\_0$. Since $\backslash tilde$ defined this way differs from $H\_0$ by a constant, we have in general
:$\backslash langle\; X\backslash rangle\_0\; =\; \backslash langle\; X\backslash rangle.$
where $\backslash langle\; X\backslash rangle$ is still the average over $\backslash tilde$, as specified above. Therefore,
:$\backslash left\backslash langle\backslash tilde\backslash right\backslash rangle\; =\; \backslash big\backslash langle\; H\_0\; +\; \backslash langle\backslash Delta\; H\backslash rangle\; \backslash big\backslash rangle\; =\; \backslash langle\; H\backslash rangle,$
and thus the inequality
:$F\; \backslash leq\; \backslash tilde$
holds. The free energy $\backslash tilde$ is the free energy of the model defined by $H\_0$ plus $\backslash langle\backslash Delta\; H\backslash rangle$. This means that
:$\backslash tilde\; =\; \backslash langle\; H\_0\backslash rangle\_0\; -\; T\; S\_0\; +\; \backslash langle\backslash Delta\; H\backslash rangle\_0\; =\; \backslash langle\; H\backslash rangle\_0\; -\; T\; S\_0,$
and thus
:$F\; \backslash leq\; \backslash langle\; H\backslash rangle\_0\; -\; T\; S\_0.$
Proof

For a classical model we can prove the Bogoliubov inequality as follows. We denote the canonical probability distributions for the Hamiltonian and the trial Hamiltonian by $P\_$ and $\backslash tilde\_$, respectively. From Gibbs' inequality we know that: :$\backslash sum\_\; \backslash tilde\_\backslash log\backslash left(\backslash tilde\_\backslash right)\backslash geq\; \backslash sum\_\; \backslash tilde\_\backslash log\backslash left(P\_\backslash right)\; \backslash ,$ holds. To see this, consider the difference between the left hand side and the right hand side. We can write this as: :$\backslash sum\_\; \backslash tilde\_\backslash log\backslash left(\backslash frac\backslash right)\; \backslash ,$ Since :$\backslash log\backslash left(x\backslash right)\backslash geq\; 1\; -\; \backslash frac\backslash ,$ it follows that: :$\backslash sum\_\; \backslash tilde\_\backslash log\backslash left(\backslash frac\backslash right)\backslash geq\; \backslash sum\_\backslash left(\backslash tilde\_\; -\; P\_\backslash right)\; =\; 0\; \backslash ,$ where in the last step we have used that both probability distributions are normalized to 1. We can write the inequality as: :$\backslash left\backslash langle\backslash log\backslash left(\backslash tilde\_\backslash right)\backslash right\backslash rangle\backslash geq\; \backslash left\backslash langle\backslash log\backslash left(P\_\backslash right)\backslash right\backslash rangle\backslash ,$ where the averages are taken with respect to $\backslash tilde\_$. If we now substitute in here the expressions for the probability distributions: :$P\_=\backslash frac\backslash ,$ and :$\backslash tilde\_=\backslash frac\backslash ,$ we get: :$\backslash left\backslash langle\; -\backslash beta\; \backslash tilde\; -\; \backslash log\backslash left(\backslash tilde\backslash right)\backslash right\backslash rangle\backslash geq\; \backslash left\backslash langle\; -\backslash beta\; H\; -\; \backslash log\backslash left(Z\backslash right)\backslash right\backslash rangle$ Since the averages of $H$ and $\backslash tilde$ are, by assumption, identical we have: :$F\backslash leq\backslash tilde$ Here we have used that the partition functions are constants with respect to taking averages and that the free energy is proportional to minus the logarithm of the partition function. We can easily generalize this proof to the case of quantum mechanical models. We denote the eigenstates of $\backslash tilde$ by $\backslash left,\; r\backslash right\backslash rangle$. We denote the diagonal components of the density matrices for the canonical distributions for $H$ and $\backslash tilde$ in this basis as: :$P\_=\backslash left\backslash langle\; r\backslash left,\; \backslash frac\backslash r\backslash right\backslash rangle\backslash ,$ and :$\backslash tilde\_=\backslash left\backslash langle\; r\backslash left,\; \backslash frac\backslash r\backslash right\backslash rangle=\backslash frac\backslash ,$ where the $\backslash tilde\_$ are the eigenvalues of $\backslash tilde$ We assume again that the averages of H and $\backslash tilde$ in the canonical ensemble defined by $\backslash tilde$ are the same: :$\backslash left\backslash langle\backslash tilde\backslash right\backslash rangle\; =\; \backslash left\backslash langle\; H\backslash right\backslash rangle\; \backslash ,$ where :$\backslash left\backslash langle\; H\backslash right\backslash rangle\; =\; \backslash sum\_\backslash tilde\_\backslash left\backslash langle\; r\backslash left,\; H\backslash r\backslash right\backslash rangle\backslash ,$ The inequality :$\backslash sum\_\; \backslash tilde\_\backslash log\backslash left(\backslash tilde\_\backslash right)\backslash geq\; \backslash sum\_\; \backslash tilde\_\backslash log\backslash left(P\_\backslash right)\; \backslash ,$ still holds as both the $P\_$ and the $\backslash tilde\_$ sum to 1. On the l.h.s. we can replace: :$\backslash log\backslash left(\backslash tilde\_\backslash right)=\; -\backslash beta\; \backslash tilde\_\; -\; \backslash log\backslash left(\backslash tilde\backslash right)\backslash ,$ On the right-hand side we can use the inequality :$\backslash left\backslash langle\backslash exp\backslash left(X\backslash right)\backslash right\backslash rangle\_\backslash geq\backslash exp\backslash left(\backslash left\backslash langle\; X\backslash right\backslash rangle\_\backslash right)\backslash ,$ where we have introduced the notation :$\backslash left\backslash langle\; Y\backslash right\backslash rangle\_\backslash equiv\backslash left\backslash langle\; r\backslash left,\; Y\backslash r\backslash right\backslash rangle\backslash ,$ for the expectation value of the operator Y in the state r. See here for a proof. Taking the logarithm of this inequality gives: :$\backslash log\backslash left;\; href="/html/ALL/s/left\backslash langle\backslash exp\backslash left(X\backslash right)\backslash right\backslash rangle\_\backslash right.html"\; ;"title="left\backslash langle\backslash exp\backslash left(X\backslash right)\backslash right\backslash rangle\_\backslash right">left\backslash langle\backslash exp\backslash left(X\backslash right)\backslash right\backslash rangle\_\backslash right$ This allows us to write: :$\backslash log\backslash left(P\_\backslash right)=\backslash log\backslash left;\; href="/html/ALL/s/left\backslash langle\backslash exp\backslash left(-\backslash beta\_H\_-\_\backslash log\backslash left(Z\backslash right)\backslash right)\backslash right\backslash rangle\_\backslash right.html"\; ;"title="left\backslash langle\backslash exp\backslash left(-\backslash beta\; H\; -\; \backslash log\backslash left(Z\backslash right)\backslash right)\backslash right\backslash rangle\_\backslash right">left\backslash langle\backslash exp\backslash left(-\backslash beta\; H\; -\; \backslash log\backslash left(Z\backslash right)\backslash right)\backslash right\backslash rangle\_\backslash right$ The fact that the averages of H and $\backslash tilde$ are the same then leads to the same conclusion as in the classical case: :$F\backslash leq\backslash tilde$Generalized Helmholtz energy

In the more general case, the mechanical term $p\backslash mathrmV$ must be replaced by the product of volume, stress, and an infinitesimal strain: :$\backslash mathrmF\; =\; V\; \backslash sum\_\; \backslash sigma\_\backslash ,\backslash mathrm\; \backslash varepsilon\_\; -\; S\backslash ,\backslash mathrmT\; +\; \backslash sum\_i\; \backslash mu\_i\; \backslash ,\backslash mathrmN\_i,$ where $\backslash sigma\_$ is the stress tensor, and $\backslash varepsilon\_$ is the strain tensor. In the case of linear elastic materials that obeyHooke's law
Hooke's law is a law of physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related enti ...

, the stress is related to the strain by
:$\backslash sigma\_\; =\; C\_\backslash varepsilon\_,$
where we are now using Einstein notation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

for the tensors, in which repeated indices in a product are summed. We may integrate the expression for $\backslash mathrmF$ to obtain the Helmholtz energy:
:$\backslash begin\; F\; \&=\; \backslash fracVC\_\backslash varepsilon\_\backslash varepsilon\_\; -\; ST\; +\; \backslash sum\_i\; \backslash mu\_i\; N\_i\; \backslash \backslash \; \&=\; \backslash fracV\backslash sigma\_\backslash varepsilon\_\; -\; ST\; +\; \backslash sum\_i\; \backslash mu\_i\; N\_i.\; \backslash end$
Application to fundamental equations of state

The Helmholtz free energy function for a pure substance (together with its partial derivatives) can be used to determine all other thermodynamic properties for the substance. See, for example, the equations of state forwater
Water (chemical formula H2O) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known li ...

, as given by the IAPWS The International Association for the Properties of Water and Steam (IAPWS) is an international non-profit association of national organizations concerned with the properties of water
Water (chemical formula H2O) is an , transparent, taste ...

in theiIAPWS-95

release.

Application to training auto-encoders

Hinton and Zemel "derive an objective function for training auto-encoder based on theminimum description length Minimum description length (MDL) refers to various formalizations of Occam's razor based on formal languages used to parsimoniously describe data. MDL is an important concept in information theory and computational learning theory.
In its most basi ...

(MDL) principle". "The description length of an input vector using a particular code is the sum of the code cost and reconstruction cost. hey
Hey or Hey! may refer to:
Music
* Hey (band), a Polish rock band
Albums
* Hey (Andreas Bourani album), ''Hey'' (Andreas Bourani album) or the title song (see below), 2014
* Hey! (Julio Iglesias album), ''Hey!'' (Julio Iglesias album) or the ti ...

define this to be the energy of the code, for reasons that will become clear later. Given an input vector, hey
Hey or Hey! may refer to:
Music
* Hey (band), a Polish rock band
Albums
* Hey (Andreas Bourani album), ''Hey'' (Andreas Bourani album) or the title song (see below), 2014
* Hey! (Julio Iglesias album), ''Hey!'' (Julio Iglesias album) or the ti ...

define the energy of a code to be the sum of the code cost and the reconstruction cost." The true expected combined cost is
:$F\; =\; \backslash sum\_i\; p\_i\; E\_i\; -\; H,$
"which has exactly the form of Helmholtz free energy".
See also

*Gibbs free energy
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qua ...

and thermodynamic free energy
The thermodynamic free energy is a concept useful in the thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properti ...

for thermodynamics history overview and discussion of ''free energy''
* Grand potential
The grand potential is a quantity used in statistical mechanics
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Space ...

* Enthalpy
Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant p ...

* Statistical mechanics
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and forc ...

* This page details the Helmholtz energy from the point of view of thermal
A thermal column (or thermal) is a column of rising air
The atmosphere of Earth is the layer of gas
Gas is one of the four fundamental states of matter
In physics
Physics is the natural science that studies matter, its ...

and statistical physics
Statistical physics is a branch of physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ...

.
* Bennett acceptance ratio
The Bennett acceptance ratio method (BAR) is an algorithm for estimating the difference in free energy between two systems (usually the systems will be simulated on the computer).
It was suggested by Charles H. Bennett (computer scientist) , Charle ...

for an efficient way to calculate free energy differences and comparison with other methods.
References

Further reading

* Atkins' ''Physical Chemistry'', 7th edition, byPeter Atkins
Peter William Atkins (born 10 August 1940) is an English chemist
A chemist (from Greek ''chēm(ía)'' alchemy; replacing ''chymist'' from Medieval Latin
Medieval Latin was the form of Latin
Latin (, or , ) is a classical language ...

and Julio de Paula, Oxford University Press
* HyperPhysics Helmholtz Free EnergHelmholtz and Gibbs Free Energies

{{Authority control Physical quantities Hermann von Helmholtz State functions Thermodynamic free energy