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In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the "useful" work obtainable from a closed thermodynamic system at a constant temperature and volume. The negative of the difference in the Helmholtz energy is equal to maximum amount of work that the system can perform in a thermodynamic process in which volume is held constant. If the volume is not held constant, part of this work will be performed as boundary work. The Helmholtz energy is commonly used for systems held at constant volume. Since in this case no work is performed on the environment, the drop in the Helmholtz energy is equal to the maximum amount of useful work that can be extracted from the system. For a system at constant temperature and volume, the Helmholtz energy is minimized at equilibrium.

The concept of Helmholtz free energy was developed by Hermann von Helmholtz, a German physician and physicist, and first presented in 1882 in a lecture called "On the thermodynamics of chemical processes".[1] It is usually denoted by the letter A  (from the German "Arbeit" or work) or the letter F. The IUPAC recommends the letter A, as well as the use of name Helmholtz energy.[2] In physics, the letter F can also be used to denote the Helmholtz energy, as Helmholtz energy is sometimes referred to as the Helmholtz function, Helmholtz free energy, or simply free energy (not to be confused with Gibbs free energy or free enthalpy).

While Gibbs free energy is most commonly used as a measure of thermodynamic potential, especially in the field of chemistry, it is inconvenient for some applications that do not 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 equations of state of pure substances.

Contents

DefinitionEdit

The Helmholtz energy is defined as[3]

 

where

A is the Helmholtz free energy (SI: joules, CGS: ergs),
U is the internal energy of the system (SI: joules, CGS: ergs),
T is the absolute temperature (kelvins) 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 transform of the internal energy U, in which temperature replaces entropy as the independent variable.

Mathematical developmentEdit

From the first law of thermodynamics in a closed system we have

 

where   is the internal energy,   is the energy added as heat, and   is the work done on the system. From the second law of thermodynamics, for a reversible process we may say that  . Also, in case of a reversible change, the work done can be expressed as   (ignoring electrical and other non-PV work):

 

Applying the product rule for differentiation to d(TS) = T dS + S dT, we have

 

and

 

The definition of A = UTS enables to rewrite this as

 

Because A is a thermodynamic function of state, 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.[4]

Minimum free energy and maximum work principlesEdit

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 (uPT processes) can also be analyzed[4] based on the fundamental thermodynamic relation 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  , the entropy increase  , and the total amount of work that can be extracted, performed by the system,  , are well defined quantities. Conservation of energy implies

 

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

 

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

 

The total entropy change is thus given by

 

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:

 

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

 

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

 

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 dA = −S dTP dV, as keeping T and V constant seems to imply dA = 0, and hence A = constant. In reality there is no contradiction: In a simple one-component system, to which the validity of the equation dA = −S dTP dV 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 Nj of particles of each type j. The differential of the free energy then generalizes to

 

where the   are the numbers of particles of type j, and the   are the corresponding chemical potentials. This equation is then again valid for both reversible and non-reversible uPT[4] 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

 

Here the   are the external variables, and the   the corresponding generalized forces.

Relation to the canonical partition functionEdit

A system kept at constant volume, temperature, and particle number is described by the canonical ensemble. The probability to find the system in some energy eigenstate r is given by

 

where

 
 
 

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:

 

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

 

The thermal average of this can be written as

 

Suppose that the system has one external variable  . Then changing the system's temperature parameter by   and the external variable by   will lead to a change in  :

 

If we write   as

 

we get

 

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

 

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

 

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

 

where c is some constant. The value of c can be determined by considering the limit T → 0. In this limit the entropy becomes  , where   is the ground-state degeneracy. The partition function in this limit is  , where   is the ground-state energy. Thus, we see that   and that

 

Relating free energy to other variablesEdit

Combining the definition of Helmholtz free energy

 

along with the fundamental thermodynamic relation

 

one can find expressions for entropy, pressure and chemical potential:[5]

 

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

 

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

 

Bogoliubov inequalityEdit

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   of the model by a trial Hamiltonian  , 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

 

where both averages are taken with respect to the canonical distribution defined by the trial Hamiltonian  , then

 

where   is the free energy of the original Hamiltonian, and   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

 

where   is exactly solvable, then we can apply the above inequality by defining

 

Here we have defined   to be the average of X over the canonical ensemble defined by  . Since   defined this way differs from   by a constant, we have in general

 

Therefore,

 

and thus the inequality

 

holds. The free energy   is the free energy of the model defined by   plus  . This means that

 

and thus

 

ProofEdit

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   and  , respectively. The inequality:

 

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

 

Since

 

it follows that:

 

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

We can write the inequality as:

 

where the averages are taken with respect to  . If we now substitute in here the expressions for the probability distributions:

 

and

 

we get:

 

Since the averages of   and   are, by assumption, identical we have:

 

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   by  . We denote the diagonal components of the density matrices for the canonical distributions for   and   in this basis as:

 

and

 

where the   are the eigenvalues of  

We assume again that the averages of H and   in the canonical ensemble defined by   are the same:

 

where

 

The inequality

 

still holds as both the   and the   sum to 1. On the l.h.s. we can replace:

 

On the right-hand side we can use the inequality

 

where we have introduced the notation

 

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   are the same then leads to the same conclusion as in the classical case:

 

Generalized Helmholtz energyEdit

In the more general case, the mechanical term (p dV) must be replaced by the product of volume, stress, and an infinitesimal strain:[6]

 

where   is the stress tensor, and   is the strain tensor. In the case of linear elastic materials that obey Hooke's law, the stress is related to the strain by

 

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   to obtain the Helmholtz energy:

 

Application to fundamental equations of stateEdit

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 IAPWS-95 release.

Application to training autoencodersEdit

Hinton and Zelem[7] "derive an objective function for training autoencoders 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, for reasons that will become clear later. 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

 

"which has exactly the form of Helmholtz free energy".

See alsoEdit

ReferencesEdit

  1. ^ von Helmholtz, H. (1882). Physical memoirs, selected and translated from foreign sources. Taylor & Francis. 
  2. ^ Gold Book. IUPAC. doi:10.1351/goldbook. Retrieved 2012-08-19. 
  3. ^ Levine, Ira. N. (1978). "Phisical Chemistry" McGraw-Hill: University of Brooklyn.
  4. ^ a b c Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes". J. Chem. Educ. 91: 402–409. Bibcode:2014JChEd..91..402S. doi:10.1021/ed3008704. 
  5. ^ "4.3 Entropy, Helmholtz Free Energy and the Partition Function". theory.physics.manchester.ac.uk. Retrieved 2016-12-06. 
  6. ^ Landau, L. D.; Lifshitz, E. M. (1986). Theory of Elasticity (Course of Theoretical Physics Volume 7). (Translated from Russian by J. B. Sykes and W. H. Reid) (Third ed.). Boston, MA: Butterworth Heinemann. ISBN 0-7506-2633-X. 
  7. ^ Hinton, G. E.; Zemel, R. S. (1994). "Autoencoders, minimum description length and Helmholtz free energy". Advances in Neural Information Processing Systems: 3–10. 

Further readingEdit

  • Atkins' Physical Chemistry, 7th edition, by Peter Atkins and Julio de Paula, Oxford University Press
  • HyperPhysics Helmholtz Free Energy [1]