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OR:

In
thermodynamics 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 th ...
, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal ...
obtainable from a closed
thermodynamic system A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundings may include other thermodynamic systems, or physical systems that a ...
at a constant
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 ...
(
isothermal In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and ...
). 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, 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 ...
or free enthalpy is most commonly used as a measure of thermodynamic potential (especially in
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
) 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 if released suddenly, usually accompanied by the production of light, heat, sound, and pressure. An expl ...
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 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 and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associat ...
, 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 working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
(IUPAC) recommends the symbol ''A'' and the name ''Helmholtz energy''. In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, the symbol ''F'' is also used in reference to ''free energy'' or ''Helmholtz function''.

# Definition

The Helmholtz free energy is defined as $F \equiv U - TS,$ where * ''F'' is the Helmholtz free energy (sometimes also called ''A'', particularly in the field of
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
) ( SI:
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
s, CGS:
erg The erg is a unit of energy equal to 10−7joules (100 nJ). It originated in the Centimetre–gram–second system of units (CGS). It has the symbol ''erg''. The erg is not an SI unit. Its name is derived from (), a Greek word meaning 'work' o ...
s), * ''U'' is the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
of the system (SI: joules, CGS: ergs), * ''T'' is the absolute temperature (
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
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 thermodyna ...
of the system (SI: joules per kelvin, CGS: ergs per kelvin). The Helmholtz energy is the Legendre transformation of the internal energy ''U'', in which temperature replaces entropy as the independent variable.

# Formal development

The
first law of thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amo ...
in a closed system provides :$\mathrmU = \delta Q\ + \delta W,$ where $U$ is the internal energy, $\delta Q$ is the energy added as heat, and $\delta W$ is the work done on the system. The
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unless ...
for a reversible process yields $\delta Q = T\,\mathrmS$. In case of a reversible change, the work done can be expressed as $\delta W = -p\,\mathrmV$ (ignoring electrical and other non-''PV'' work) and so: :$\mathrmU = T\,\mathrmS - p\,\mathrmV.$ Applying the product rule for differentiation to $\mathrm\left(TS\right) = T \mathrmS\, + S\mathrmT$, it follows :$\mathrmU = \mathrm\left(TS\right) - S\,\mathrmT - p\,\mathrmV,$ and :$\mathrm\left(U - TS\right) = -S\,\mathrmT - p\,\mathrmV.$ The definition of $F = U - TS$ enables to rewrite this as :$\mathrmF = -S\,\mathrmT - p\,\mathrmV.$ Because ''F'' is a thermodynamic function of state, this relation is also valid for a process (without electrical work or composition change) that is not reversible.

# Minimum free energy and maximum work principles

The laws of thermodynamics are only directly applicable to systems in thermal equilibrium. If we wish to describe phenomena like chemical reactions, then the best we can do is 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 $\Delta U$, 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 thermodyna ...
increase $\Delta S$, and the total amount of work that can be extracted, performed by the system, $W$, are well defined quantities. Conservation of energy implies :$\Delta U_\text + \Delta U + W = 0.$ 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 :$Q_\text = \Delta U_\text = -\left(\Delta U + W\right).$ 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 :$\Delta S_\text = \frac = -\frac.$ The total entropy change is thus given by :$\Delta S_\text + \Delta S = -\frac.$ 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: :$\Delta S_\text + \Delta S = -\frac.$ Since the total change in entropy must always be larger or equal to zero, we obtain the inequality :$W \leq -\Delta F.$ 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 :$\Delta F \leq 0,$ 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''''j'' of particles of each type ''j''. The differential of the free energy then generalizes to :$dF = -S\,dT - P\,dV + \sum_j \mu_j\,dN_j,$ where the $N_$ are the numbers of particles of type j and the $\mu_$ are the corresponding
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 ...
s. This equation is then again valid for both reversible and non-reversible changes. In case of a spontaneous change at constant T and V, the last term will thus be negative. In case there are other external parameters, the above relation further generalizes to :$dF = -S\,dT - \sum_i X_i\,dx_i + \sum_j \mu_j\,dN_j.$ Here the $x_i$ are the external variables, and the $X_i$ the corresponding
generalized forces Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, Fi, i=1,..., n, acting on a system A system is a group of interacting or interrelate ...
.

# Relation to the canonical partition function

A system kept at constant volume, temperature, and particle number is described by the canonical ensemble. The probability of finding the system in some energy eigenstate ''r'', for any microstate ''i'', is given by $P_r = \frac,$ where *$\beta = \frac,$ *$E_r$ is the energy of accessible state $r$ *$Z = \sum_i e^.$ ''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: :$U \equiv \langle E \rangle = \sum_r P_r E_r = \sum_r \frac = \sum_r \frac = \frac = -\frac.$ If the system is in state ''r'', then the generalized force corresponding to an external variable ''x'' is given by :$X_r = -\frac.$ The thermal average of this can be written as :$X = \sum_r P_r X_r = \frac \frac.$ Suppose that the system has one external variable $x$. Then changing the system's temperature parameter by $d\beta$ and the external variable by $dx$ will lead to a change in $\log Z$: :$d\left(\log Z\right) = \frac\,d\beta + \frac\,dx = -U\,d\beta + \beta X\,dx.$ If we write $U\,d\beta$ as :$U\,d\beta = d\left(\beta U\right) - \beta\, dU,$ we get :$d\left(\log Z\right) = -d\left(\beta U\right) + \beta\, dU + \beta X \,dx.$ This means that the change in the internal energy is given by :$dU = \frac\,d\left(\log Z + \beta U\right) - X\,dx.$ In the thermodynamic limit, the fundamental thermodynamic relation should hold: :$dU = T\, dS - X\, dx.$ This then implies that the entropy of the system is given by :$S = k\log Z + \frac + c,$ where ''c'' is some constant. The value of ''c'' can be determined by considering the limit ''T'' → 0. In this limit the entropy becomes $S = k \log \Omega_0$, where $\Omega_0$ is the ground-state degeneracy. The partition function in this limit is $\Omega_0 e^$, where $U_0$ is the ground-state energy. Thus, we see that $c = 0$ and that :$F = -kT\log Z.$

## Relating free energy to other variables

Combining the definition of Helmholtz free energy :$F = U - T S$ along with the fundamental thermodynamic relation $dF = -S\,dT - P\,dV + \mu\,dN,$ one can find expressions for entropy, pressure and chemical potential: :$S = \left.-\left\left( \frac \right\right) \_, \quad P = \left.-\left\left( \frac \right\right) \_, \quad \mu = \left.\left\left( \frac \right\right) \_.$ These three equations, along with the free energy in terms of the partition function, :$F = -kT\log Z,$ 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 do Legendre transformations for different systems. For example, for a system with a magnetic field or potential, it is true that :$m = \left.-\left\left( \frac \right\right) \_, \quad V = \left.\left \left( \frac \right\right) \_.$

# Bogoliubov inequality

Computing the free energy is an intractable problem for all but the simplest models in statistical physics. A powerful approximation method is
mean-field theory In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random ( stochastic) models by studying a simpler model that approximates the original by averaging over degrees of ...
, 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 $\tilde$, 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 :$\left\langle\tilde\right\rangle = \langle H \rangle,$ where both averages are taken with respect to the canonical distribution defined by the trial Hamiltonian $\tilde$, then the Bogoliubov inequality states :$F \leq \tilde,$ where $F$ is the free energy of the original Hamiltonian, and $\tilde$ is the free energy of the trial Hamiltonian. We will prove this below. 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 applied in the following way. If we write the Hamiltonian as :$H = H_0 + \Delta H,$ where $H_0$ is some exactly solvable Hamiltonian, then we can apply the above inequality by defining :$\tilde = H_0 + \langle\Delta H\rangle_0.$ Here we have defined $\langle X\rangle_0$ to be the average of ''X'' over the canonical ensemble defined by $H_0$. Since $\tilde$ defined this way differs from $H_0$ by a constant, we have in general :$\langle X\rangle_0 = \langle X\rangle.$ where $\langle X\rangle$ is still the average over $\tilde$, as specified above. Therefore, :$\left\langle\tilde\right\rangle = \big\langle H_0 + \langle\Delta H\rangle \big\rangle = \langle H\rangle,$ and thus the inequality :$F \leq \tilde$ holds. The free energy $\tilde$ is the free energy of the model defined by $H_0$ plus $\langle\Delta H\rangle$. This means that :$\tilde = \langle H_0\rangle_0 - T S_0 + \langle\Delta H\rangle_0 = \langle H\rangle_0 - T S_0,$ and thus :$F \leq \langle H\rangle_0 - T S_0.$

## Proof of the Bogoliubov inequality

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 $\tilde_$, respectively. From Gibbs' inequality we know that: :$\sum_ \tilde_\log\left\left(\tilde_\right\right)\geq \sum_ \tilde_\log\left\left(P_\right\right) \,$ holds. To see this, consider the difference between the left hand side and the right hand side. We can write this as: :$\sum_ \tilde_\log\left\left(\frac\right\right) \,$ Since :$\log\left\left(x\right\right)\geq 1 - \frac\,$ it follows that: :$\sum_ \tilde_\log\left\left(\frac\right\right)\geq \sum_\left\left(\tilde_ - P_\right\right) = 0 \,$ where in the last step we have used that both probability distributions are normalized to 1. We can write the inequality as: :$\left\langle\log\left\left(\tilde_\right\right)\right\rangle\geq \left\langle\log\left\left(P_\right\right)\right\rangle\,$ where the averages are taken with respect to $\tilde_$. If we now substitute in here the expressions for the probability distributions: :$P_=\frac\,$ and :$\tilde_=\frac\,$ we get: :$\left\langle -\beta \tilde - \log\left\left(\tilde\right\right)\right\rangle\geq \left\langle -\beta H - \log\left\left(Z\right\right)\right\rangle$ Since the averages of $H$ and $\tilde$ are, by assumption, identical we have: :$F\leq\tilde$ 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 $\tilde$ by $\left, r\right\rangle$. We denote the diagonal components of the density matrices for the canonical distributions for $H$ and $\tilde$ in this basis as: :$P_=\left\langle r\left, \frac\r\right\rangle\,$ and :$\tilde_=\left\langle r\left, \frac\r\right\rangle=\frac\,$ where the $\tilde_$ are the eigenvalues of $\tilde$ We assume again that the averages of H and $\tilde$ in the canonical ensemble defined by $\tilde$ are the same: :$\left\langle\tilde\right\rangle = \left\langle H\right\rangle \,$ where :$\left\langle H\right\rangle = \sum_\tilde_\left\langle r\left, H\r\right\rangle\,$ The inequality :$\sum_ \tilde_\log\left\left(\tilde_\right\right)\geq \sum_ \tilde_\log\left\left(P_\right\right) \,$ still holds as both the $P_$ and the $\tilde_$ sum to 1. On the l.h.s. we can replace: :$\log\left\left(\tilde_\right\right)= -\beta \tilde_ - \log\left\left(\tilde\right\right)\,$ On the right-hand side we can use the inequality :$\left\langle\exp\left\left(X\right\right)\right\rangle_\geq\exp\left\left(\left\langle X\right\rangle_\right\right)\,$ where we have introduced the notation :$\left\langle Y\right\rangle_\equiv\left\langle r\left, Y\r\right\rangle\,$ for the expectation value of the operator Y in the state r. See here for a proof. Taking the logarithm of this inequality gives: : This allows us to write: : The fact that the averages of H and $\tilde$ are the same then leads to the same conclusion as in the classical case: :$F\leq\tilde$

# Generalized Helmholtz energy

In the more general case, the mechanical term $p\mathrmV$ must be replaced by the product of volume, stress, and an infinitesimal strain: :$\mathrmF = V \sum_ \sigma_\,\mathrm \varepsilon_ - S\,\mathrmT + \sum_i \mu_i \,\mathrmN_i,$ where $\sigma_$ is the stress tensor, and $\varepsilon_$ is the strain tensor. In the case of linear elastic materials that obey
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
, the stress is related to the strain by :$\sigma_ = C_\varepsilon_,$ where we are now using
Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
for the tensors, in which repeated indices in a product are summed. We may integrate the expression for $\mathrmF$ to obtain the Helmholtz energy: :$\begin F &= \fracVC_\varepsilon_\varepsilon_ - ST + \sum_i \mu_i N_i \\ &= \fracV\sigma_\varepsilon_ - ST + \sum_i \mu_i N_i. \end$

# 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 for
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts a ...
, 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 and steam, particularly thermophysical properties and other aspe ...
in thei
IAPWS-95
release.

# Application to training auto-encoders

Hinton and Zemel "derive an objective function for training auto-encoder based on the
minimum description length Minimum Description Length (MDL) is a model selection principle where the shortest description of the data is the best model. MDL methods learn through a data compression perspective and are sometimes described as mathematical applications of Occam ...
(MDL) principle". "The description length of an input vector using a particular code is the sum of the code cost and reconstruction cost. heydefine this to be the energy of the code, for reasons that will become clear later. Given an input vector, heydefine the energy of a code to be the sum of the code cost and the reconstruction cost." The true expected combined cost is :$F = \sum_i p_i E_i - H,$ "which has exactly the form of Helmholtz free energy".

*
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 ...
and
thermodynamic free energy The thermodynamic free energy is a concept useful in the thermodynamics of chemical or thermal processes in engineering and science. The change in the free energy is the maximum amount of work that a thermodynamic system can perform in a process ...
for thermodynamics history overview and discussion of ''free energy'' * Grand potential *
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 ...
*
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 b ...
* This page details the Helmholtz energy from the point of view of
thermal A thermal column (or thermal) is a rising mass of buoyant air, a convective current in the atmosphere, that transfers heat energy vertically. Thermals are created by the uneven heating of Earth's surface from solar radiation, and are an exampl ...
and
statistical physics Statistical physics is a branch of physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical ...
. * Bennett acceptance ratio for an efficient way to calculate free energy differences and comparison with other methods.