11.03.04 · stat-mech-physics / ensembles

Isothermal-Isobaric (NPT) Ensemble: Gibbs Free Energy as the Natural Potential

shipped3 tiersLean: none

Anchor (Master): Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (1985), Ch. 9–10; Landau & Lifshitz, Statistical Physics, Part 1, 3rd ed. (1980), Sec. 31

Intuition Beginner

The canonical ensemble 11.03.01 describes a system at fixed temperature $T$ and fixed volume $V$ . The system exchanges energy with a heat bath, and the Helmholtz free energy $F$ governs equilibrium. But most experiments — especially in chemistry — happen in open containers at atmospheric pressure, not in rigid boxes. A reaction in a beaker, a protein folding in solution, a gas expanding against the atmosphere: all of these occur at constant temperature and constant pressure.

The isothermal-isobaric ensemble (also called the NPT ensemble) is built for this situation. The system has fixed particle number $N$ , fixed temperature $T$ , and fixed pressure $P$ . It exchanges energy with a heat bath (like the canonical ensemble) and its volume can change (unlike the canonical ensemble). A piston that can move, in contact with a heat bath at temperature $T$ and under constant external pressure $P$ , is the prototypical NPT system.

The central quantity is the Gibbs free energy $G$ . Just as the Helmholtz free energy $F = - k T ln Z$ governs the canonical ensemble, the Gibbs free energy $G = - k T ln Δ$ governs the NPT ensemble. The symbol $Δ$ (sometimes written $Δ_{N}$ or $Y$ ) is the isothermal-isobaric partition function. It collects all the Boltzmann weights $e^{- (E + P V) / k T}$ for every possible energy $E$ and every possible volume $V$ the system might have.

The Gibbs free energy has a direct physical meaning: $G$ is the maximum non-expansion work (sometimes called "useful work") that can be extracted from a system at constant temperature and pressure. When a chemical reaction releases energy, only the part captured as $- Δ G$ (the change in Gibbs free energy) is available to do useful work — the rest goes to $P V$ work against the atmosphere.

Three relations connect $G$ to other thermodynamic potentials. The Gibbs free energy equals the Helmholtz free energy plus $P V$ : $G = F + P V$ . It also equals the enthalpy minus $T S$ : $G = H - T S$ . And from the Euler relation for a single-component system, $G = μ N$ : the Gibbs free energy is the chemical potential times the number of particles.

Visual Beginner

Picture a cylinder fitted with a frictionless piston. The piston is exposed to the atmosphere on one side, exerting a constant pressure $P$ . Inside the cylinder is a gas. The walls conduct heat, connecting the gas to a large heat bath at temperature $T$ .

The piston moves: the volume fluctuates. At any instant the gas might occupy $V = 1.00$ L, or $V = 1.02$ L, or $V = 0.99$ L. The average volume $⟨ V ⟩$ is set by the equilibrium between the gas pressure and the external pressure $P$ . The fluctuations $Δ V$ are tiny for macroscopic systems but become significant for nanoscale systems.

A histogram of observed volumes shows a bell-shaped curve centred at $⟨ V ⟩$ . The width of the curve measures how compressible the substance is: a stiff solid has a narrow distribution (small fluctuations), while a compressible gas has a wider distribution.

Worked example Beginner

Consider an ideal gas with $N$ particles at temperature $T$ and pressure $P$ . At a given volume $V$ , the canonical partition function (the quantity that counts up all Boltzmann-weighted states at that volume) is $Z = (V / λ^{3})^{N} / N!$ where $λ = h / 2 π mk T$ is the thermal wavelength.

The NPT partition function $Δ$ collects contributions from every possible volume $V$ , each weighted by $e^{- P V / k T}$ :

Δ = \frac{1}{N ! λ ^{3 N}} \times \frac{N !}{( P / k T ) ^{N + 1}} = \frac{( k T ) ^{N + 1}}{λ ^{3 N} P ^{N + 1}} .

The second factor comes from evaluating a standard mathematical function (the gamma function) that arises when you add up all the $V^{N} e^{- P V / k T}$ terms from $V = 0$ to $V = \infty$ .

The Gibbs free energy is $G = - k T ln Δ$ :

G = - k T [(N + 1) ln (k T) - 3 N ln λ - (N + 1) ln P] .

For large $N$ , the $+ 1$ is negligible:

G \approx N k T [ln P - ln (k T / λ^{3})] = N k T ln (\frac{P λ ^{3}}{k T}) .

Using the ideal gas law $P V = N k T$ (so $P / k T = N / V$ ), this becomes $G = N k T ln (n λ^{3})$ where $n = N / V$ is the number density. This is consistent with $G = μ N$ and $μ = k T ln (n λ^{3})$ , as expected.

Check your understanding Beginner

Formal definition Intermediate+

The isothermal-isobaric ensemble describes a system at fixed particle number $N$ , fixed temperature $T$ , and fixed pressure $P$ . The system is coupled to both a heat bath (fixing $T$ ) and a mechanism that maintains constant pressure (fixing $P$ ). Both energy $E$ and volume $V$ fluctuate.

The isothermal-isobaric partition function is

Δ (N, P, T) = \int_{0}^{\infty} e^{- β P V} Z (N, V, T) d V,

where $β = 1/ (k_{B} T)$ and $Z (N, V, T)$ is the canonical partition function 11.03.01. Equivalently, writing the sum over microstates directly:

Δ = \int_{0}^{\infty} i \sum e^{- β (E_{i} + P V)} d V,

where the sum runs over all microstates $i$ at volume $V$ with energy $E_{i}$ .

The Gibbs free energy is

G (N, P, T) = - k_{B} T ln Δ.

Probability distribution over volume

The probability density for the system to have volume $V$ is

P (V) = \frac{e ^{- β P V} Z ( N , V , T )}{Δ} .

The mean volume and other thermodynamic quantities follow from derivatives of $G$ :

⟨ V ⟩ = - (\frac{\partial G}{\partial P})_{T, N}, S = - (\frac{\partial G}{\partial T})_{P, N}, μ = (\frac{\partial G}{\partial N})_{T, P} .

Legendre-transform structure

The Gibbs free energy is the Legendre transform of the Helmholtz free energy with respect to volume 11.01.02:

G (T, P, N) = F (T, V, N) + P V, where P = - (\frac{\partial F}{\partial V})_{T, N} .

The natural variables of $G$ are $(T, P, N)$ , replacing $(T, V, N)$ for $F$ . This is the Legendre structure introduced in 11.01.02: replacing a mechanical variable ( $V$ ) with its conjugate thermodynamic force ( $P$ ). The three equivalent forms are

G = F + P V = H - T S = μ N .

Volume fluctuations

The variance of the volume is

(Δ V)^{2} = ⟨ V^{2} ⟩ - ⟨ V ⟩^{2} = - k_{B} T (\frac{\partial ⟨ V ⟩}{\partial P})_{T, N} = k_{B} T V κ_{T},

where $κ_{T} = - \frac{1}{V} (\frac{\partial V}{\partial P})_{T}$ is the isothermal compressibility. For an ideal gas, $κ_{T} = 1/ P$ , giving

\frac{( Δ V ) ^{2}}{⟨ V ⟩ ^{2}} = \frac{k T}{P ⟨ V ⟩} = \frac{1}{N} .

The relative volume fluctuation $Δ V / ⟨ V ⟩ = 1/ N$ vanishes in the thermodynamic limit, establishing the equivalence of NPT and NVT ensembles for macroscopic systems.

Key derivation: Ideal gas in the NPT ensemble Intermediate+

Proposition. For an ideal gas of $N$ particles with mass $m$ in the NPT ensemble at pressure $P$ and temperature $T$ , the Gibbs free energy is

G = - N k T ln (\frac{k T}{P λ ^{3}}) + k T ln (\frac{k T}{P}),

where $λ = h / 2 π mk T$ . In the thermodynamic limit, the second term is negligible and $G \approx N k T ln (P λ^{3} / k T) = - N k T ln (V / (N λ^{3}))$ .

Proof. Start from the canonical partition function for the ideal gas:

Z (N, V, T) = \frac{1}{N !} (\frac{V}{λ ^{3}})^{N} .

The NPT partition function is

Δ = \int_{0}^{\infty} e^{- β P V} Z (N, V, T) d V = \frac{1}{N ! λ ^{3 N}} \int_{0}^{\infty} V^{N} e^{- β P V} d V .

The integral is the gamma function $Γ (N + 1) / (β P)^{N + 1} = N! / (β P)^{N + 1}$ . Substituting:

Δ = \frac{1}{N ! λ ^{3 N}} \cdot \frac{N !}{( β P ) ^{N + 1}} = \frac{( k T ) ^{N + 1}}{λ ^{3 N} P ^{N + 1}} .

Taking the logarithm:

ln Δ = (N + 1) ln (k T) - 3 N ln λ - (N + 1) ln P .

So $G = - k T ln Δ = (N + 1) k T ln P - (N + 1) k T ln (k T) + 3 N k T ln λ$ . Using $λ = h / 2 π mk T$ , this yields the standard ideal-gas Gibbs free energy. For $N ≫ 1$ , $N + 1 \approx N$ and

G \approx N k T ln (\frac{P λ ^{3}}{k T}) = N k T ln (n λ^{3}) = μ N,

confirming $G = μ N$ with $μ = k T ln (n λ^{3})$ . $□$

Bridge. The ideal gas in the NPT ensemble demonstrates the Legendre duality between $F (T, V, N)$ and $G (T, P, N)$ : the $P V$ term in $G = F + P V$ appears naturally as the Boltzmann weight $e^{- β P V}$ in the partition function. This is the same Legendre structure introduced in 11.01.02, now realised as an integral transform over volume rather than an algebraic substitution. The NPT partition function is the Laplace transform (in $V$ ) of the canonical partition function, just as the grand canonical partition function 11.03.02 is the $z$ -transform (in $N$ ) of the canonical partition function.

Exercises Intermediate+

Exercise 7 (hard, symbolic).

Consider a one-dimensional polymer of $N$ segments each of length $a$ , modelled as a Gaussian chain. The canonical partition function at fixed end-to-end extension $L$ is $Z (N, L, T) \propto e^{- L^{2} / (2 N a^{2})}$ . Compute the Gibbs-like free energy $G = - k T ln Δ$ where $Δ = \int_{0}^{\infty} e^{- β f L} Z d L$ and extract the mean extension $⟨ L ⟩$ . Here $f$ is the applied force (the one-dimensional analogue of pressure).

Hint

This is the one-dimensional analogue of the NPT ensemble, with force $f$ replacing pressure and extension $L$ replacing volume. Complete the square in the exponent and evaluate the Gaussian integral.

Answer

Δ \propto \int_{0}^{\infty} e^{- β f L} e^{- L^{2} / (2 N a^{2})} d L .

Completing the square in the exponent:

- β f L - \frac{L ^{2}}{2 N a ^{2}} = - \frac{1}{2 N a ^{2}} (L - N f a^{2} β)^{2} + \frac{N f ^{2} a ^{2} β ^{2}}{2} .

Δ \propto e^{N f^{2} a^{2} β^{2} /2} \int_{0}^{\infty} e^{- (L - N f a^{2} β)^{2} / (2 N a^{2})} d L .

For large $N$ , the Gaussian is sharply peaked at $L^{*} = N f a^{2} / (k T)$ and the integral is approximately $2 π N a^{2}$ . So:

ln Δ \approx \frac{N f ^{2} a ^{2}}{2 ( k T ) ^{2}} + \frac{1}{2} ln (2 π N a^{2}) .

G = - k T ln Δ \approx - \frac{N f ^{2} a ^{2}}{2 k T} - \frac{k T}{2} ln (2 π N a^{2}) .

⟨ L ⟩ = - \frac{\partial G}{\partial f} = \frac{N f a ^{2}}{k T} .

This is the Hookean response of a polymer: the mean extension is proportional to the applied force with effective spring constant $k T / (N a^{2})$ — an entropic elasticity arising from the statistics of the chain conformations.

Exercise 8 (hard, symbolic).

Show that the NPT ensemble is equivalent to the canonical ensemble in the thermodynamic limit by proving that the probability distribution $P (V)$ is sharply peaked around $⟨ V ⟩$ with relative width $Δ V / ⟨ V ⟩ \sim 1/ N$ .

Hint

Write $P (V) = e^{f (V)}$ where $f (V) = ln Z (N, V, T) - β P V - ln Δ$ . Expand $f (V)$ around its maximum at $V = V^{*}$ to second order and evaluate the Gaussian width.

Answer

Define $f (V) = ln Z (N, V, T) - β P V - ln Δ$ . The maximum occurs at $V^{*}$ where $f^{'} (V^{*}) = 0$ :

\frac{\partial ln Z}{\partial V}_{V^{*}} = β P .

Since $\partial ln Z / \partial V = β P_{int}$ where $P_{int}$ is the internal pressure, the condition is $P_{int} (V^{*}) = P$ , confirming $V^{*}$ is the equilibrium volume.

Expand $f (V) \approx f (V^{*}) + \frac{1}{2} f^{''} (V^{*}) (V - V^{*})^{2}$ where

f^{''} (V^{*}) = \frac{\partial ^{2} ln Z}{\partial V ^{2}}_{V^{*}} = β \frac{\partial P _{int}}{\partial V}_{V^{*}} = - \frac{β}{V ^{*} κ _{T}} .

The distribution is Gaussian with variance $(Δ V)^{2} = 1/∣ f^{''} (V^{*}) ∣ = V^{*} κ_{T} / (k T)$ . From the fluctuation relation $(Δ V)^{2} = k T V^{*} κ_{T}$ :

\frac{Δ V}{V ^{*}} = \frac{k T κ _{T}}{V ^{*}} .

For an ideal gas, $κ_{T} = 1/ P$ and $P V^{*} = N k T$ , so $k T κ_{T} / V^{*} = 1/ N$ and $Δ V / V^{*} = 1/ N \to 0$ as $N \to \infty$ . In the thermodynamic limit, the volume is sharply determined and the NPT and NVT ensembles give identical thermodynamics.

Lean formalization Intermediate+

The isothermal-isobaric ensemble has no Mathlib formalization. The NPT partition function $Δ (N, P, T) = \int_{0}^{\infty} e^{- β P V} Z (N, V, T) d V$ is a Laplace transform of the canonical partition function with respect to volume. Formalising this requires: (1) a probability density on the continuous volume domain $V \in (0, \infty)$ proportional to $e^{- β P V} Z (N, V, T)$ ; (2) the identification $G = - k T ln Δ$ ; (3) the Legendre-transform duality $G = F + P V$ ; (4) the fluctuation relation $(Δ V)^{2} = k T V κ_{T}$ . The first item is the load-bearing prerequisite, building on the canonical probability measure established for the NVT ensemble 11.03.01.

Advanced results Master

Thermodynamic integration for free energy computation

The NPT ensemble is the workhorse for computing Gibbs free energies in molecular simulation. The key technique is thermodynamic integration: if $G (T, P)$ is known for a reference system (typically an ideal gas), the Gibbs free energy of a target system is obtained by integrating the derivative along a path connecting the two.

Consider a system with Hamiltonian $H (λ) = (1 - λ) H_{ref} + λ H_{target}$ where $λ \in [0, 1]$ interpolates between the reference and target. Then:

G_{target} - G_{ref} = \int_{0}^{1} ⟨ \frac{\partial H}{\partial λ} ⟩_{λ} d λ = \int_{0}^{1} ⟨ H_{target} - H_{ref} ⟩_{λ} d λ,

where $⟨ \cdot ⟩_{λ}$ denotes the NPT average at coupling parameter $λ$ . The integral is evaluated by running simulations at several values of $λ$ and computing the mean energy difference at each. This method converges faster than direct free-energy estimation from $Δ$ because it avoids the need to sample the tails of the distribution.

A related technique is free-energy perturbation (Zwanzig, 1954):

G_{target} - G_{ref} = - k T ln ⟨ e^{- β (H_{target} - H_{ref})} ⟩_{ref} .

This works well when the two systems are similar; for large differences, thermodynamic integration is preferred because it breaks the path into small steps where the exponential averaging is well-behaved.

G(T, P) as a generating function

The Gibbs free energy $G (T, P, N)$ is the generating function for thermodynamic quantities through its first and second derivatives:

V = - \frac{\partial G}{\partial P}, S = - \frac{\partial G}{\partial T}, μ = \frac{\partial G}{\partial N},

κ_{T} = - \frac{1}{V} \frac{\partial ^{2} G}{\partial P ^{2}}, C_{P} = - T \frac{\partial ^{2} G}{\partial T ^{2}}, α = \frac{1}{V} \frac{\partial ^{2} G}{\partial T \partial P},

where $α$ is the thermal expansion coefficient. The Maxwell relations follow from the symmetry of mixed second derivatives: $\partial^{2} G / \partial T \partial P = \partial^{2} G / \partial P \partial T$ , giving $- (\partial S / \partial P)_{T} = (\partial V / \partial T)_{P}$ .

This generating-function perspective unifies thermodynamics: all measurable quantities are encoded in a single function $G (T, P, N)$ , and all relations between them follow from differentiation. The relationship between $G$ and $Δ$ is the same as that between $F$ and $Z$ in the canonical ensemble: the free energy is $- k T$ times the logarithm of the partition function, and all thermodynamics follows from the derivatives of the free energy.

NPT molecular dynamics: barostats

In molecular dynamics, the NPT ensemble is realised by coupling the system to both a thermostat (to fix $T$ ) and a barostat (to fix $P$ ). The most common barostats are:

Berendsen barostat (1984): scales coordinates at each step by a factor $λ = (1 - β_{P} Δ t / τ_{P})^{1/3}$ where $τ_{P}$ is the coupling time constant and $β_{P}$ is related to the compressibility. Simple to implement but does not generate the exact NPT distribution — it produces a time-rescaling rather than sampling from the correct Boltzmann distribution.
Andersen barostat (1980): treats the volume as a dynamical variable coupled to the system through an extended Lagrangian. The volume evolves according to an equation of motion driven by the difference between the internal and external pressure. Generates the correct NPT distribution.
Parrinello-Rahman barostat (1981): extends the Andersen barostat to allow changes in the simulation box shape as well as size. The box vectors become dynamical variables. Essential for studying structural phase transitions where the crystal symmetry changes.
Nosé-Hoover barostat (combined with a Nosé-Hoover thermostat): uses extended-system methods to couple both temperature and pressure. Produces the exact NPT distribution for sufficiently small timesteps and is the preferred choice for production calculations of Gibbs free energies.

Synthesis. The NPT ensemble completes the standard ensemble framework by providing the natural description for constant-pressure conditions, which are the most common experimental situation in chemistry and materials science. The Gibbs free energy $G (T, P, N)$ is the thermodynamic potential conjugate to the Helmholtz free energy $F (T, V, N)$ via the Legendre transform with respect to volume 11.01.02. The NPT partition function $Δ$ is the Laplace transform (over $V$ ) of the canonical partition function, just as the grand canonical partition function $Ξ$ 11.03.02 is the $z$ -transform (over $N$ ) of the canonical partition function. The three transforms — Legendre in thermodynamics, Laplace in statistics, $z$ -transform in grand canonical — are three faces of the same mathematical structure: replacing a fixed extensive variable by its conjugate intensive variable and summing (or integrating) over the fluctuations.

Full proof set Master

Proposition. The NPT partition function $Δ$ is the Laplace transform of the canonical partition function with respect to volume, and the Gibbs free energy $G = - k T ln Δ$ satisfies all the standard thermodynamic relations.

Proof. Define $f (β P) := ln Δ = ln \int_{0}^{\infty} e^{- β P V} Z (N, V, T) d V$ . The first derivative gives the mean volume:

f^{'} (β P) = \frac{\int _{0}^{\infty} ( - V ) e ^{- β P V} Z d V}{Δ} = - ⟨ V ⟩ .

Since $G = - k T f$ , we have $⟨ V ⟩ = - (\partial G / \partial P)_{T, N}$ , confirming the thermodynamic relation.

The second derivative gives the volume fluctuation:

f^{''} (β P) = ⟨ V^{2} ⟩ - ⟨ V ⟩^{2} = (Δ V)^{2} .

From $G = - k T f$ : $\partial^{2} G / \partial P^{2} = - k T f^{''} (β P) \cdot β^{2} = - β (Δ V)^{2}$ . Since $(Δ V)^{2} = k T V κ_{T}$ , we recover $\partial^{2} G / \partial P^{2} = - β \cdot k T V κ_{T} = - V κ_{T}$ , consistent with $\partial^{2} G / \partial P^{2} = \partial (- V) / \partial P = - \partial V / \partial P = V κ_{T}$ .

The entropy follows from the $T$ -derivative. From $G = - k T ln Δ$ :

S = - \frac{\partial G}{\partial T}_{P, N} = k ln Δ + k T \frac{\partial ln Δ}{\partial T}_{P, N} .

The first term is $- G / T$ . The second term, carrying through the $T$ -dependence in both $β$ and $Z$ , gives $⟨ E ⟩ / T + P ⟨ V ⟩ / T = H / T$ . So $S = - G / T + H / T = (H - G) / T$ , confirming $G = H - T S$ . $□$

Connections Master

11.01.02 The Legendre transform $G = F + P V$ is the concrete realisation of the abstract Legendre structure; the NPT ensemble provides the statistical-mechanical underpinning for replacing $V$ with $P$ as the independent variable.
11.03.01 The canonical ensemble is the parent ensemble: $Δ$ integrates $Z$ over volume with the Boltzmann weight $e^{- β P V}$ .
11.03.02 The grand canonical ensemble performs a similar transform but in particle number rather than volume; the two can be combined to give the $(μ, P, T)$ ensemble.
11.04.01 The general theory of ensembles and their equivalence in the thermodynamic limit underpins the NPT framework.
11.06.01 Phase transitions at constant pressure are described by non-analyticities in $G (T, P)$ ; coexistence of phases occurs when $G$ is equal for two phases at the same $T$ and $P$ .

Historical and philosophical context Master

Josiah Willard Gibbs introduced the concept of thermodynamic potentials — including the Gibbs free energy — in his monumental paper "On the Equilibrium of Heterogeneous Substances" (1875--1878). Gibbs recognised that different experimental conditions require different thermodynamic potentials as their natural descriptions, and that these potentials are related by Legendre transforms. The potential now called $G$ (sometimes denoted $Φ$ in early literature, and still called the "Gibbs function" in some traditions) was identified by Gibbs as the relevant quantity for systems at constant temperature and pressure — the conditions of most chemical experiments.

The statistical-mechanical formulation of the NPT ensemble followed from Gibbs' 1902 Elementary Principles in Statistical Mechanics, where he introduced the general framework of ensembles. The isothermal-isobaric ensemble is obtained by applying a Laplace transform (over volume) to the canonical ensemble, paralleling the $z$ -transform over particle number that yields the grand canonical ensemble. The mathematical unity of these transforms — Legendre in thermodynamics, Laplace and $z$ -transform in statistical mechanics — is a structural insight that Gibbs himself understood but that was made explicit only by later authors.

E. A. Guggenheim, in his Thermodynamics: An Advanced Treatment for Chemists and Physicists (first edition 1949, fifth edition 1967), provided a systematic treatment of thermodynamic potentials and their Legendre relationships. Guggenheim emphasised the Gibbs free energy as the fundamental quantity for chemical thermodynamics, establishing the notation and conventions still used today. His work clarified the connection between the Legendre-transform structure of thermodynamics and the statistical-mechanical ensemble formalism, and his textbook became the standard reference for the rigorous treatment of thermodynamic potentials.

The development of molecular dynamics methods in the late 20th century — particularly the Andersen barostat (1980), Parrinello-Rahman barostat (1981), and Nosé-Hoover methods — made the NPT ensemble a practical tool for computational chemistry. These algorithms enable direct simulation of systems at constant pressure, allowing computation of Gibbs free energies, phase equilibria, and equation-of-state data from first principles. The NPT ensemble is now the default choice for most biomolecular simulations, where systems are naturally at constant atmospheric pressure.

Bibliography Master

@book{gibbs1902,
  author = {Gibbs, J. Willard},
  title = {Elementary Principles in Statistical Mechanics},
  publisher = {Yale University Press},
  year = {1902}
}

@book{pathria-beale2011,
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  title = {Statistical Mechanics},
  edition = {3rd},
  publisher = {Academic Press},
  year = {2011}
}

@book{reif1965,
  author = {Reif, Frederick},
  title = {Fundamentals of Statistical and Thermal Physics},
  publisher = {McGraw-Hill},
  year = {1965}
}

@book{callen1985,
  author = {Callen, Herbert B.},
  title = {Thermodynamics and an Introduction to Thermostatistics},
  edition = {2nd},
  publisher = {Wiley},
  year = {1985}
}

@book{guggenheim1967,
  author = {Guggenheim, E. A.},
  title = {Thermodynamics: An Advanced Treatment for Chemists and Physicists},
  edition = {5th},
  publisher = {North-Holland},
  year = {1967}
}

@book{landau-lifshitz1980,
  author = {Landau, L. D. and Lifshitz, E. M.},
  title = {Statistical Physics, Part 1},
  edition = {3rd},
  publisher = {Pergamon Press},
  year = {1980}
}

@article{andersen1980,
  author = {Andersen, Hans C.},
  title = {Molecular dynamics simulations at constant pressure and/or temperature},
  journal = {Journal of Chemical Physics},
  volume = {72},
  pages = {2384--2393},
  year = {1980}
}

@article{parrinello-rahman1981,
  author = {Parrinello, M. and Rahman, A.},
  title = {Polymorphic transitions in single crystals: A new molecular dynamics method},
  journal = {Journal of Applied Physics},
  volume = {52},
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  year = {1981}
}

@article{zwanzig1954,
  author = {Zwanzig, Robert W.},
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}

Prerequisites

11.03.01
11.04.01
11.01.02

Tier anchors

beginner: Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (1985), Ch. 6; Schroeder, An Introduction to Thermal Physics (2000), Ch. 5
intermediate: Reif, Fundamentals of Statistical and Thermal Physics (1965), Ch. 6; Pathria & Beale, Statistical Mechanics, 3rd ed. (2011), Ch. 4
master: Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (1985), Ch. 9–10; Landau & Lifshitz, Statistical Physics, Part 1, 3rd ed. (1980), Sec. 31

References

Gibbs — Elementary Principles in Statistical Mechanics (Yale, 1902) · Ch. IV The canonical distribution. Gibbs formulated the ensemble framework that generalises to the isothermal-isobaric ensemble through Legendre transformation of the volume variable.
Pathria & Beale — Statistical Mechanics, 3rd ed. (Academic Press, 2011) · Ch. 4 Other ensembles and equivalence in the thermodynamic limit. The isothermal-isobaric partition function Delta and the identification G = -kT ln Delta.
Reif — Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965) · Ch. 6 The canonical and grand canonical distributions. Derivation of the NPT ensemble by coupling to both a heat bath and a pressure reservoir.
Callen — Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, 1985) · Ch. 6 The thermodynamic potentials and the Legendre transform; Ch. 9–10 Statistical mechanics and the connection to thermodynamic potentials via Legendre structure.
Guggenheim — Thermodynamics: An Advanced Treatment for Chemists and Physicists (North-Holland, 1967) · Ch. 1–4 Systematic treatment of thermodynamic potentials and the Gibbs free energy as the natural potential for constant T, P conditions.

Estimated time

beginner: 15 min
intermediate: 40 min
master: 70 min