Isothermal–isobaric ensemble explained

The isothermal–isobaric ensemble (constant temperature and constant pressure ensemble) is a statistical mechanical ensemble that maintains constant temperature

T and constant pressure P applied. It is also called the NpT -ensemble, where the number of particles N is also kept as a constant. This ensemble plays an important role in chemistry as chemical reactions are usually carried out under constant pressure condition.^[1] The NPT ensemble is also useful for measuring the equation of state of model systems whose virial expansion for pressure cannot be evaluated, or systems near first-order phase transitions.^[2]

In the ensemble, the probability of a microstate

i is Z ^{-1} e ^{-\beta(E(i)} , where Z is the partition function, E(i) is the internal energy of the system in microstate i , and V(i) is the volume of the system in microstate i .

The probability of a macrostate is

Z ^{-1} e ^{-\beta(E} = Z ^{-1} e ^{-\beta} , where G is the Gibbs free energy.

Derivation of key properties

The partition function for the

NpT -ensemble can be derived from statistical mechanics by beginning with a system of N identical atoms described by a Hamiltonian of the form p ^{2/2m+U(r} ^{n)} and contained within a box of volume V = L ^{3} . This system is described by the partition function of the canonical ensemble in 3 dimensions:Z ^{sys} (N, V, T) = (1)/(Λ ^{3N} N!) \int ^{L} _{0} ... \int ^{L} _{0} d r ^{N} \exp( - \beta U( r ^{N))} ,

where

Λ = \sqrt{h ^{2} \beta/( 2 \pi m)} , the thermal de Broglie wavelength (\beta =1 /k _{B} T and k _{B} is the Boltzmann constant), and the factor 1 /N! (which accounts for indistinguishability of particles) both ensure normalization of entropy in the quasi-classical limit.^[2] It is convenient to adopt a new set of coordinates defined by L s _{i} = r _{i} such that the partition function becomesZ ^{sys} (N, V, T) = (V ^{N})/(Λ ^{3N} N!) \int ^{1} _{0} ... \int ^{1} _{0} d s ^{N} \exp( - \beta U( s ^{N))} .

If this system is then brought into contact with a bath of volume

V _{0} at constant temperature and pressure containing an ideal gas with total particle number M such that M - N \gg N , the partition function of the whole system is simply the product of the partition functions of the subsystems:Z ^{sys+bath} (N, V, T) = (V ^{M - N} _{0- V)})/(Λ ^{3M} N!(M - N)!) \int d s ^{M-N} \int d s ^{N} \exp( - \beta U( s ^{N))} .

The integral over the

s ^{M-N} coordinates is simply 1 . In the limit that V _{0} → inf ty , M → inf ty while (M - N)/V _{0=\rho} stays constant, a change in volume of the system under study will not change the pressure p of the whole system. Taking V/V _{0} → 0 allows for the approximation (V ^{M - N} _{0- V)} = V ^{M - N} _{0} ( 1- V/V ^{M - N} _{0 )} ≈ V ^{M - N} _{0} \exp( - (M - N)V/V _{0)} . For an ideal gas, (M - N)/V _{0} = \rho = \beta P gives a relationship between density and pressure. Substituting this into the above expression for the partition function, multiplying by a factor \beta P (see below for justification for this step), and integrating over the volume V then gives\Delta ^{sys+bath} (N, P, T) = (\beta P V ^{M - N} _{0})/(Λ ^{3M} N!(M - N)!) \int dV V ^{N} \exp({ - \beta P V}) \int d s ^{N} \exp( - \beta U( s )) .

The partition function for the bath is simply

\Delta ^{bath} = V ^{M - N} _{0} /[(M - N)! Λ ^{3(M-N)} . Separating this term out of the overall expression gives the partition function for the NpT -ensemble:\Delta ^{sys} (N, P, T) = (\beta P)/(Λ ^{3N} N!) \int dV V ^{N} \exp( - \beta P V) \int d s ^{N} \exp( - \beta U( s )) .

Using the above definition of

Z ^{sys} (N,V,T) , the partition function can be rewritten as\Delta ^{sys} (N, P, T) = \beta P \int dV \exp( - \beta P V) Z ^{sys} (N, V, T) ,

which can be written more generally as a weighted sum over the partition function for the canonical ensemble

\Delta(N, P, T) = \int Z(N, V, T) \exp( - \beta PV ) C dV.

The quantity

C is simply some constant with units of inverse volume, which is necessary to make the integral dimensionless. In this case, C = \beta P , but in general it can take on multiple values. The ambiguity in its choice stems from the fact that volume is not a quantity that can be counted (unlike e.g. the number of particles), and so there is no “natural metric” for the final volume integration performed in the above derivation.^[2] This problem has been addressed in multiple ways by various authors,^{[3] [4]} leading to values for C with the same units of inverse volume. The differences vanish (i.e. the choice of C becomes arbitrary) in the thermodynamic limit, where the number of particles goes to infinity.^[5]

The

NpT -ensemble can also be viewed as a special case of the Gibbs canonical ensemble, in which the macrostates of the system are defined according to external temperature T and external forces acting on the system J . Consider such a system containing N particles. The Hamiltonian of the system is then given by l{H}- J ⋅ x where l{H} is the system's Hamiltonian in the absence of external forces and x are the conjugate variables of J . The microstates \mu of the system then occur with probability defined by ^[6] p(\mu, x ) = \exp[ - \beta l{H}(\mu)+\beta J ⋅ x ]/ l{Z}

where the normalization factor

l{Z} is defined byl{Z}(N, J , T) = \sum _{\mu,x} \exp[\beta J ⋅ x - \beta l{H}(\mu)] .

This distribution is called generalized Boltzmann distribution by some authors.^[7]

The

NpT -ensemble can be found by taking J =- P and x = V . Then the normalization factor becomesl{Z}(N, J , T) = \sum _{\mu, \{ r _{i\}} \in V} \exp[ - \beta P V - \beta( p ^{2/2m+U(r} ^{N))]} ,

where the Hamiltonian has been written in terms of the particle momenta

p _{i} and positions r _{i} . This sum can be taken to an integral over both V and the microstates \mu . The measure for the latter integral is the standard measure of phase space for identical particles: rm{d} \Gamma _{N} = (1)/(h ^{3N!}) \prod ^{N} _{i =1} d ^{3 p} _{i} d ^{3 r} _{i} .^[6] The integral over \exp( - \beta p ^{2/2m)} term is a Gaussian integral, and can be evaluated explicitly as\int \prod ^{N} _{i =1} (d ^{3 p} _{i})/(h ^{3}) \exp [ - \beta \sum ^{N} _{i =1} (p ^{2} _{i})/(2 m) ] = (1)/(Λ ^{3N}) .

Inserting this result into

l{Z}(N,P,T) gives a familiar expression:l{Z}(N, P, T) = (1)/(Λ ^{3N} N!) \int dV \exp( - \beta P V) \int d r ^{N} \exp( - \beta U( r )) = \int dV \exp( - \beta P V)Z(N, V, T) .^[6]

This is almost the partition function for the

NpT -ensemble, but it has units of volume, an unavoidable consequence of taking the above sum over volumes into an integral. Restoring the constant C yields the proper result for \Delta(N, P, T) .

From the preceding analysis it is clear that the characteristic state function of this ensemble is the Gibbs free energy,

G(N, P, T) = - k _{B} T ln \Delta(N, P, T)

This thermodynamic potential is related to the Helmholtz free energy (logarithm of the canonical partition function),

F , in the following way:^[1] G = F + PV.

Applications

• Constant-pressure simulations are useful for determining the equation of state of a pure system. Monte Carlo simulations using the

NpT -ensemble are particularly useful for determining the equation of state of fluids at pressures of around 1 atm, where they can achieve accurate results with much less computational time than other ensembles.^[2]

• Zero-pressure

NpT -ensemble simulations provide a quick way of estimating vapor-liquid coexistence curves in mixed-phase systems.^[2] NpT -ensemble Monte Carlo simulations have been applied to study the excess properties^[8] and equations of state ^[9] of various models of fluid mixtures.

• The

NpT -ensemble is also useful in molecular dynamics simulations, e.g. to model the behavior of water at ambient conditions.^[10]

Notes and References

1. Book: Dill . Ken A. . Bromberg . Sarina . Stigter . Dirk . . 2003 . . New York.
2. Book: Frenkel . Daan. . Smit . Berend . . 2002 . . New York.
3. Attard . Phil . 1995. On the density of volume states in the isobaric ensemble. Journal of Chemical Physics. 103 . 24 . 9884–9885 . 10.1063/1.469956. 1995JChPh.103.9884A .
4. Koper . Ger J. M. . Reiss . Howard . 1996. Length Scale for the Constant Pressure Ensemble: Application to Small Systems and Relation to Einstein Fluctuation Theory . Journal of Physical Chemistry. 100 . 1 . 422–432 . 10.1021/jp951819f.
5. Book: Hill . Terrence . Statistical Mechanics: Principles and Selected Applications . 1987 . . New York.
6. Book: Kardar . Mehran . Statistical Physics of Particles . 2007 . . New York.
7. Gao . Xiang . Gallicchio . Emilio . Adrian . Roitberg . 2019 . The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy . The Journal of Chemical Physics. 151. 3. 034113. 10.1063/1.5111333. 31325924 . 1903.02121 . 2019JChPh.151c4113G . 118981017 .
8. McDonald . I. R. . 1972. NpT -ensemble Monte Carlo calculations for binary liquid mixtures. . 23 . 1 . 41–58 . 10.1080/00268977200100031. 1972MolPh..23...41M .
9. Wood . W. W. . 1970. NpT -Ensemble Monte Carlo Calculations for the Hard Disk Fluid. Journal of Chemical Physics . 52 . 2 . 729–741 . 10.1063/1.1673047 . 1970JChPh..52..729W .
10. Schmidt . Jochen . VandeVondele . Joost . Kuo . I. F. William . Sebastiani . Daniel . Siepmann . J. Ilja . Hutter . Jürg . Mundy . Christopher J. . 2009. Isobaric-Isothermal Molecular Dynamics Simulations Utilizing Density Functional Theory:An Assessment of the Structure and Density of Water at Near-Ambient Conditions . Journal of Physical Chemistry B . 113 . 35 . 11959–11964 . 10.1021/jp901990u. 19663399 . 980890 .