Helmholtz free energy

2025-09-18T05:20:15Z

2409:40F4:30A7:77E7:2C56:2226:3B1F:98FC:

{{Short description|Thermodynamic potential}}
{{Thermodynamics|expanded=Potentials}}

In [[thermodynamics]], the '''Helmholtz free energy''' (or '''Helmholtz energy''') is a [[thermodynamic potential]] that measures the useful [[work (thermodynamics)|work]] obtainable from a [[closed system|closed]] [[thermodynamic system]] at a constant [[temperature]] ([[Isothermal process|isothermal]]). 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]] or free enthalpy is most commonly used as a measure of thermodynamic potential (especially in [[chemistry]]) when it is convenient for applications that occur at constant ''pressure''. For example, in [[explosives]] research Helmholtz free energy is often used, since explosive reactions by their nature induce pressure changes. It is also frequently used to define fundamental [[equation of state|equations of state]] of pure substances.

The concept of free energy was developed by [[Hermann von Helmholtz]], a German physicist, and first presented in 1882 in a lecture called "On the thermodynamics of chemical processes".<ref>{{cite book | author = von Helmholtz, H. | date = 1882 | title = Physical memoirs, selected and translated from foreign sources | publisher = [[Taylor & Francis]]}}</ref> From the German word ''Arbeit'' (work), the [[International Union of Pure and Applied Chemistry]] (IUPAC) recommends the symbol ''A'' and the name ''Helmholtz energy''.<ref name=":0">{{cite book | title = Gold Book | year = 2019 | publisher = [[IUPAC]] | url = http://goldbook.iupac.org/H02772.html | access-date = 2012-08-19 | doi=10.1351/goldbook| editor1-last = Gold | editor1-first = Victor }}</ref> In [[physics]], the symbol ''F'' is also used in reference to ''free energy'' or ''Helmholtz function''.

==Definition==
The Helmholtz free energy is defined as<ref>Levine, Ira. N. (1978). "''Physical Chemistry''" McGraw-Hill: University of Brooklyn.</ref>
<math display="block">A \equiv U - TS,</math>
where
* ''A'' is the Helmholtz free energy<ref name=":0" /> (sometimes also called ''F'', particularly in the field of [[physics]]) ([[SI]]: [[joule]]s, [[Centimetre–gram–second system of units|CGS]]: [[erg]]s),
* ''U'' is the [[internal energy]] of the system (SI: joules, CGS: ergs),
* ''T'' is the absolute temperature ([[kelvin]]s) of the surroundings, modelled as a heat bath,
* ''S'' is the [[entropy]] 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]] in a closed system provides
<math display="block">\mathrm{d}U = \delta Q\ - \delta W,</math>
where <math>U</math> is the internal energy, <math>\delta Q</math> is the energy added as heat, and <math>\delta W</math> is the work done on the system. The [[second law of thermodynamics]] for a [[reversible process (thermodynamics)|reversible process]] yields <math>\delta Q = T\,\mathrm{d}S</math>. In case of a reversible change, the work done can be expressed as <math>\delta W = p\,\mathrm{d}V</math> (ignoring electrical and other non-''PV'' work) and so:
<math display="block">\mathrm{d}U = T\,\mathrm{d}S - p\,\mathrm{d}V.</math>

Applying the product rule for differentiation to <math>\mathrm{d}(TS) = T \mathrm{d}S\, + S\mathrm{d}T</math>, it follows
<math display="block">\mathrm{d}U = \mathrm{d}(TS) - S\,\mathrm{d}T - p\,\mathrm{d}V,</math>
and
<math display="block">\mathrm{d}(U - TS) = -S\,\mathrm{d}T - p\,\mathrm{d}V.</math>

The definition of <math>A = U - TS</math> allows us to rewrite this as
<math display="block">\mathrm{d}A = -S\,\mathrm{d}T - p\,\mathrm{d}V.</math>

Because ''A'' is a thermodynamic [[Functions of state|function of state]], this [[fundamental thermodynamic relation|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 <math>\Delta U</math>, the [[entropy]] increase <math>\Delta S</math>, and the total amount of work that can be extracted, performed by the system, <math>W</math>, are well defined quantities. Conservation of energy implies

<math display="block">\Delta U_\text{bath} + \Delta U + W = 0.</math>

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

<math display="block">Q_\text{bath} = \Delta U_\text{bath} = -(\Delta U + W).</math>

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

<math display="block">\Delta S_\text{bath} = \frac{Q_\text{bath}}{T} = -\frac{\Delta U + W}{T}.</math>

The total entropy change is thus given by

<math display="block">\Delta S_\text{bath} + \Delta S = -\frac{\Delta U - T\Delta S + W}{T}.</math>

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:

<math display="block">\Delta S_\text{bath} + \Delta S = -\frac{\Delta A + W}{T}.</math>

Since the total change in entropy must always be larger or equal to zero, we obtain the inequality

<math display="block">W \leq -\Delta A.</math>

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

<math display="block">\Delta A \leq 0,</math>

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 <math>\mathrm{d} A = -S \,\mathrm{d} T - p \,\mathrm{d} V</math>, as keeping ''T'' and ''V'' constant seems to imply <math>\mathrm{d} A = 0</math>, and hence <math>A = \mathrm{const.}</math> In reality there is no contradiction: In a simple one-component system, to which the validity of the equation <math>\mathrm{d} A = -S \,\mathrm{d} T - p \,\mathrm{d} V</math> is restricted, no process can occur at constant ''T'' and ''V'', since there is a unique <math>P(T, V)</math> 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''<sub>''j''</sub> of particles of each type ''j''. The differential of the free energy then generalizes to

<math display="block">\mathrm{d} A = -S \,\mathrm{d} T + P \,\mathrm{d} V + \sum_j \mu_j \,\mathrm{d} N_j,</math>

where the <math>N_{j}</math> are the numbers of particles of type j and the <math>\mu_{j}</math> are the corresponding [[chemical potential]]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 [[fundamental thermodynamic relation|relation]] further generalizes to

<math display="block">\mathrm{d} A = -S \,\mathrm{d} T - \sum_i X_i \,\mathrm{d} x_i + \sum_j \mu_j \,\mathrm{d} N_j.</math>

Here the <math>x_i</math> are the external variables, and the <math>X_i</math> the corresponding [[generalized forces]].

==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
<math display="block">P_r = \frac{e^{-\beta E_r}}{Z},</math>
where
*<math>\beta = \frac{1}{k T},</math>
*<math>E_r</math> is the energy of accessible state <math>r</math>
*<math display="inline">Z = \sum_i e^{-\beta E_i}.</math>

''Z'' is called the [[partition function (statistical mechanics)|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:

<math display="block">U \equiv \langle E \rangle
= \sum_r P_r E_r
= \sum_r \frac{e^{-\beta E_r} E_r}{Z}
= \sum_r \frac{-\frac{\partial}{\partial \beta} e^{-\beta E_r}}{Z}
= \frac{-\frac{\partial}{\partial \beta} \sum_r e^{-\beta E_r}}{Z}
= -\frac{\partial \log Z}{\partial \beta}.</math>

If the system is in state ''r'', then the generalized force corresponding to an external variable ''x'' is given by

<math display="block">X_r = -\frac{\partial E_r}{\partial x}.</math>

The thermal average of this can be written as

<math display="block">X = \sum_r P_r X_r = \frac{1}{\beta} \frac{\partial \log Z}{\partial x}.</math>

Suppose that the system has one external variable <math>x</math>. Then changing the system's temperature parameter by <math>d\beta</math> and the external variable by <math>dx</math> will lead to a change in <math>\log Z</math>:

<math display="block">d(\log Z) = \frac{\partial\log Z}{\partial\beta}\,d\beta + \frac{\partial\log Z}{\partial x}\,dx = -U\,d\beta + \beta X\,dx.</math>

If we write <math>U\,d\beta</math> as

<math display="block">U\,d\beta = d(\beta U) - \beta\, dU,</math>

we get

<math display="block">d(\log Z) = -d(\beta U) + \beta\, dU + \beta X \,dx.</math>

This means that the change in the internal energy is given by

<math display="block">dU = \frac{1}{\beta}\,d(\log Z + \beta U) - X\,dx.</math>

In the thermodynamic limit, the [[fundamental thermodynamic relation]] should hold:

<math display="block">dU = T\, dS - X\, dx.</math>

This then implies that the entropy of the system is given by

<math display="block">S = k\log Z + \frac{U}{T} + c,</math>

where ''c'' is some constant. The value of ''c'' can be determined by considering the limit ''T'' → 0. In this limit the entropy becomes <math>S = k \log \Omega_0</math>, where <math>\Omega_0</math> is the ground-state degeneracy. The partition function in this limit is <math>\Omega_0 e^{-\beta U_0}</math>, where <math>U_0</math> is the ground-state energy. Thus, we see that <math>c = 0</math> and that

{{Equation box 1
|indent=:
|title= Microscopic definition of <math>A</math>:
|equation=<math>\, A = -kT\log Z.</math>
|border = 1
}}

=== Relating free energy to other variables ===

Combining the definition of Helmholtz free energy

<math display="block">A = U - T S</math>

along with the fundamental thermodynamic relation

<math display="block">\mathrm{d} A = -S \,\mathrm{d} T - P\,\mathrm{d} V + \mu \,\mathrm{d} N,</math>

one can find expressions for entropy, pressure and chemical potential:<ref>{{Cite web|url=http://theory.physics.manchester.ac.uk/~judith/stat_therm/node70.html|title=4.3 Entropy, Helmholtz Free Energy and the Partition Function|website=theory.physics.manchester.ac.uk|access-date=2016-12-06}}</ref>

<math display="block">S = \left.-\left( \frac{\partial A}{\partial T} \right) \right|_{V,N}, \quad
P = \left.-\left( \frac{\partial A}{\partial V} \right) \right|_{T,N}, \quad
\mu = \left.\left( \frac{\partial A}{\partial N} \right) \right|_{T,V}.</math>

These three equations, along with the free energy in terms of the partition function,

<math display="block">A = -kT\log Z,</math>

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 transformation]]s for different systems. For example, for a system with a magnetic field or potential, it is true that

<math display="block">m = \left.-\left( \frac{\partial A}{\partial B} \right) \right|_{T,N}, \quad
V = \left.\left ( \frac{\partial A}{\partial Q} \right) \right|_{N,T}.</math>

==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]], which is a variational method based on the Bogoliubov inequality. This inequality can be formulated as follows.

Suppose we replace the real Hamiltonian <math>H</math> of the model by a trial Hamiltonian <math>\tilde{H}</math>, 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

<math display="block">\left\langle\tilde{H}\right\rangle = \langle H \rangle,</math>

where both averages are taken with respect to the canonical distribution defined by the trial Hamiltonian <math>\tilde{H}</math>, then the Bogoliubov inequality states

<math display="block">A \leq \tilde{A},</math>

where <math>A</math> is the free energy of the original Hamiltonian, and <math>\tilde{A}</math> 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

<math display="block">H = H_0 + \Delta H,</math>

where <math>H_0</math> is some exactly solvable Hamiltonian, then we can apply the above inequality by defining

<math display="block">\tilde{H} = H_0 + \langle\Delta H\rangle_0.</math>

Here we have defined <math>\langle X\rangle_0</math> to be the average of ''X'' over the canonical ensemble defined by <math>H_0</math>. Since <math>\tilde{H}</math> defined this way differs from <math>H_0</math> by a constant, we have in general

<math display="block">\langle X\rangle_0 = \langle X\rangle.</math>

where <math>\langle X\rangle</math> is still the average over <math>\tilde{H}</math>, as specified above. Therefore,

<math display="block">\left\langle\tilde{H}\right\rangle = \big\langle H_0 + \langle\Delta H\rangle \big\rangle = \langle H\rangle,</math>

and thus the inequality

<math display="block">A \leq \tilde{A}</math>

holds. The free energy <math>\tilde{A}</math> is the free energy of the model defined by <math>H_0</math> plus <math>\langle\Delta H\rangle</math>. This means that

<math display="block">\tilde{A} = \langle H_0\rangle_0 - T S_0 + \langle\Delta H\rangle_0 = \langle H\rangle_0 - T S_0,</math>

and thus

<math display="block">A\leq \langle H\rangle_0 - T S_0.</math>

===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 <math>P_{r}</math> and <math>\tilde{P}_{r}</math>, respectively. From [[Gibbs' inequality]] we know that:

<math display="block">\sum_{r} \tilde{P}_{r}\log\left(\tilde{P}_{r}\right)\geq \sum_{r} \tilde{P}_{r}\log\left(P_{r}\right) \,</math>

holds. To see this, consider the difference between the left hand side and the right hand side. We can write this as:

<math display="block">\sum_{r} \tilde{P}_{r}\log\left(\frac{\tilde{P}_{r}}{P_{r}}\right) \,</math>

Since

<math display="block">\log\left(x\right)\geq 1 - \frac{1}{x}\,</math>

it follows that:

<math display="block">\sum_{r} \tilde{P}_{r}\log\left(\frac{\tilde{P}_{r}}{P_{r}}\right)\geq \sum_{r}\left(\tilde{P}_{r} - P_{r}\right) = 0 \,</math>

where in the last step we have used that both probability distributions are normalized to 1.

We can write the inequality as:

<math display="block">\left\langle\log\tilde{P}_{r}\right\rangle \geq \left\langle\log P_{r} \right\rangle</math>

where the averages are taken with respect to <math>\tilde{P}_{r}</math>. If we now substitute in here the expressions for the probability distributions:

<math display="block">P_{r} = \frac{\exp\left[-\beta H(r)\right]}{Z}</math>

and

<math display="block">\tilde{P}_{r} = \frac{\exp\left[-\beta\tilde{H}(r)\right]}{\tilde{Z}}</math>

we get:

<math display="block">\left\langle -\beta \tilde{H} - \log \tilde{Z} \right\rangle\geq \left\langle -\beta H - \log Z \right\rangle</math>

Since the averages of <math>H</math> and <math>\tilde{H}</math> are, by assumption, identical we have:

<math display="block">A\leq\tilde{A}
</math>

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 <math>\tilde{H}</math> by <math>\left|r\right\rangle</math>. We denote the diagonal components of the density matrices for the canonical distributions for <math>H</math> and <math>\tilde{H}</math> in this basis as:

<math display="block">P_{r}=\left\langle r\left|\frac{\exp\left[-\beta H\right]}{Z}\right|r\right\rangle\,</math>

and

<math display="block">\tilde{P}_{r}=\left\langle r\left|\frac{\exp\left[-\beta\tilde{H}\right]}{\tilde{Z}}\right|r\right\rangle=\frac{\exp\left(-\beta\tilde{E}_{r}\right)}{\tilde{Z}}\,</math>

where the <math>\tilde{E}_{r}</math> are the eigenvalues of <math>\tilde{H}</math>

We assume again that the averages of H and <math>\tilde{H}</math> in the canonical ensemble defined by <math>\tilde{H}</math> are the same:

<math display="block">\left\langle\tilde{H}\right\rangle = \left\langle H\right\rangle \,</math>

where
<math display="block">\left\langle H\right\rangle = \sum_{r}\tilde{P}_{r}\left\langle r\left|H\right|r\right\rangle\,</math>

The inequality

<math display="block">\sum_{r} \tilde{P}_{r} \log \tilde{P}_{r} \geq \sum_{r} \tilde{P}_{r} \log P_{r}</math>

still holds as both the <math>P_{r}</math> and the <math>\tilde{P}_{r}</math> sum to 1. On the left-hand side we can replace:

<math display="block">\log \tilde{P}_{r} = -\beta \tilde{E}_{r} - \log \tilde{Z}</math>

On the right-hand side we can use the inequality

<math display="block">\left\langle e^X \right\rangle_{r} \geq e^{{\left\langle X \right\rangle}_{r}}</math>
where we have introduced the notation

<math display="block">\left\langle Y\right\rangle_{r}\equiv\left\langle r\left|Y\right|r\right\rangle\,</math>

for the expectation value of the operator Y in the state r. [[Jensen's inequality#Statistical physics|See here]] for a proof. Taking the logarithm of this inequality gives:

<math display="block">\log\left[\left\langle e^X \right\rangle_{r}\right]\geq\left\langle X\right\rangle_{r}\,</math>

This allows us to write:

<math display="block">\log P_{r} = \log\left[\left\langle \exp\left(-\beta H - \log Z \right)\right\rangle_{r}\right] \geq \left\langle -\beta H - \log Z\right\rangle_{r}</math>

The fact that the averages of H and <math>\tilde{H}</math> are the same then leads to the same conclusion as in the classical case:

<math display="block">A\leq\tilde{A}</math>

== Generalized Helmholtz energy ==

In the more general case, the mechanical term <math>p\mathrm{d}V</math> must be replaced by the product of volume, [[Stress (physics)|stress]], and an infinitesimal strain:<ref>{{cite book
| last = Landau
| first = L. D.
| author-link = Lev Landau
|author2=Lifshitz, E. M. |author-link2=Evgeny Lifshitz
| others = (Translated from Russian by J. B. Sykes and W. H. Reid)
| year = 1986
| title = Theory of Elasticity (Course of Theoretical Physics Volume 7)
| edition = Third
| publisher = Butterworth Heinemann
| location = Boston, MA
| isbn = 0-7506-2633-X
}}</ref>

<math display="block">\mathrm{d}A = V \sum_{ij} \sigma_{ij}\,\mathrm{d} \varepsilon_{ij} - S\,\mathrm{d}T + \sum_i \mu_i \,\mathrm{d}N_i,</math>

where <math>\sigma_{ij}</math> is the stress tensor, and <math>\varepsilon_{ij}</math> is the strain tensor. In the case of linear [[Elasticity (physics)|elastic]] materials that obey [[Hooke's law]], the stress is related to the strain by

<math display="block">\sigma_{ij} = C_{ijkl}\varepsilon_{kl},</math>

where we are now using [[Einstein notation]] for the tensors, in which repeated indices in a product are summed. We may integrate the expression for <math>\mathrm{d}A</math> to obtain the Helmholtz energy:

<math display="block">
\begin{align}
A &= \frac{1}{2}VC_{ijkl}\varepsilon_{ij}\varepsilon_{kl} - ST + \sum_i \mu_i N_i \\
&= \frac{1}{2}V\sigma_{ij}\varepsilon_{ij} - ST + \sum_i \mu_i N_i.
\end{align}
</math>

== 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]], as given by the [[IAPWS]] in their [https://web.archive.org/web/20110726024808/http://www.iapws.org/relguide/IAPWS95-Rev.pdf IAPWS-95] release.

== Application to training auto-encoders ==

Hinton and Zemel<ref>{{cite journal|last1=Hinton|first1=G. E.|last2=Zemel|first2=R. S.|title=Autoencoders, minimum description length and Helmholtz free energy|journal=Advances in Neural Information Processing Systems|date=1994|pages=3–10|url=https://proceedings.neurips.cc/paper/1993/file/9e3cfc48eccf81a0d57663e129aef3cb-Paper.pdf}}</ref> "derive an objective function for training [[autoencoder|auto-encoder]] based on the [[minimum description length]] (MDL) principle". "The description length of an input vector using a particular code is the sum of the code cost and reconstruction cost. They define this to be the energy of the code. Given an input vector, they define the energy of a code to be the sum of the code cost and the reconstruction cost." The true expected combined cost is
<math display="block">A = \sum_i p_i E_i - H ,</math>
"which has exactly the form of Helmholtz free energy".

== See also ==
* [[Gibbs free energy]] and [[thermodynamic free energy]] for thermodynamics history overview and discussion of ''free energy''
* [[Grand potential]]
* [[Enthalpy]]
* [[Statistical mechanics]]
* This page details the Helmholtz energy from the point of view of [[thermodynamics|thermal]] and [[statistical physics]].
* [[Bennett acceptance ratio]] for an efficient way to calculate free energy differences and comparison with other methods.

== References ==
{{Reflist}}

== Further reading ==
* Atkins' ''Physical Chemistry'', 7th edition, by [[Peter Atkins]] and Julio de Paula, Oxford University Press
* HyperPhysics Helmholtz Free Energy [http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/helmholtz.html Helmholtz and Gibbs Free Energies]

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Helmholtz free energy