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45

Ensembles

Field: Statistical Mechanics

A collection of systems used in statistical mechanics to describe the thermodynamic properties of a system.

Sequence of Expressions

Let H:(ρ,t)→R\mathcal{H}: (\boldsymbol{\rho}, t) \rightarrow \mathbb{R} be the Hamiltonian of the system, defined on the phase space Γ⊂R6N\Gamma \subset \mathbb{R}^{6N}. The system is constrained to the energy shell ΩE,ϵ={ρ∈Γ∣E≤H(ρ)≤E+ϵ}\Omega_{E, \epsilon} = \left\{ \boldsymbol{\rho} \in \Gamma \mid E \le \mathcal{H}(\boldsymbol{\rho}) \le E + \epsilon \right\}. The number of accessible microstates, W(E,V,N)W(E, V, N), is given by the phase space volume integral: W(E,V,N)=∫ΩE,ϵd3Nqh3NN!δ(H(q,p)−E)⋅1ϵdE′d3NpW(E, V, N) = \int_{\Omega_{E, \epsilon}} \frac{d^{3N} \boldsymbol{q}}{h^{3N} N!} \delta\left(\mathcal{H}(\boldsymbol{q}, \boldsymbol{p}) - E\right) \cdot \frac{1}{\epsilon} dE' d^{3N} \boldsymbol{p} where q=(q1,…,qN)\boldsymbol{q} = (q_1, \dots, q_N) and p=(p1,…,pN)\boldsymbol{p} = (p_1, \dots, p_N) are the generalized coordinates and momenta, respectively. The entropy SS is then defined via the Boltzmann relation: S(E,V,N)=kBln⁡W(E,V,N)S(E, V, N) = k_B \ln W(E, V, N) where kBk_B is the Boltzmann constant.
Let H:Phase Space→R\mathcal{H}: \text{Phase Space} \to \mathbb{R} be the Hamiltonian of the system. Define the inverse temperature β=1/(kBT)\beta = 1/(k_B T). The Canonical Partition Function ZZ for a system with fixed volume VV and particle number NN is given by the integral over the phase space Γ\Gamma: Z(β,V,N)=∫Γe−βH(p,q)dpdqZ(\beta, V, N) = \int_{\Gamma} e^{-\beta \mathcal{H}(\mathbf{p}, \mathbf{q})} d\boldsymbol{p} d\boldsymbol{q} where dpdqd\boldsymbol{p} d\boldsymbol{q} is the measure on the phase space. The probability density ρ(p,q)\rho(\mathbf{p}, \mathbf{q}) of finding the system in a microstate (p,q)(\mathbf{p}, \mathbf{q}) is the Boltzmann distribution: ρ(p,q)=e−βH(p,q)Z(β,V,N)\rho(\mathbf{p}, \mathbf{q}) = \frac{e^{-\beta \mathcal{H}(\mathbf{p}, \mathbf{q})}}{Z(\beta, V, N)} Furthermore, the Helmholtz Free Energy AA is derived from the partition function via the Legendre transform relationship: A(β,V,N)=−kBTln⁡Z(β,V,N)A(\beta, V, N) = -k_B T \ln Z(\beta, V, N) This ensemble characterizes the statistical mechanical description of systems governed by the fixed parameters (β,V,N)(\beta, V, N).
Let S\mathcal{S} be the system, and R\mathcal{R} be the reservoir. The system is characterized by a Hamiltonian H(x,N)\mathcal{H}(\mathbf{x}, N) and a fixed chemical potential μ\mu and temperature TT. The Grand Canonical Partition Function Z\mathcal{Z} is defined over the state space Ωstates\Omega_{\text{states}} and particle number NN: \n\n$$\mathcal{Z}(\mu, T, V) = \sum_{N=0}^{\infty} \frac{z^N}{N!} \text{Tr}\\text{e}\left(e^{-\beta \mathcal{H}_N}\right) = \text{Tr}\\text{e}\left(e^{-\beta (\mathcal{H} - \mu\hat{N})\right)$$\n\nwhere $\beta = 1/(k_B T)$, $z = e^{\beta \mu}$ is the fugacity, $\mathcal{H}$ is the system Hamiltonian operator, and $\hat{N}$ is the particle number operator. The Grand Potential $\Omega$ is then derived via the thermodynamic relation:\n\n$$\Omega(\mu, T, V) = -k_B T \ln \mathcal{Z}(\mu, T, V)$$\n\nFurthermore, the expectation value of any observable $\mathcal{O}$ is given by:\n\n$$\langle \mathcal{O} \rangle = \frac{1}{\mathcal{Z}} \text{Tr}\\text{e}\left(\mathcal{O} e^{-\beta (\mathcal{H} - \mu\hat{N})}\right)$$
Let S\mathcal{S} be the discrete set of accessible microstates of a system, and let EiE_i be the energy eigenvalue associated with state i∈Si \in \mathcal{S}. Define the inverse temperature β\beta as β=1/(kT)\beta = 1/(kT), where kk is the Boltzmann constant and TT is the absolute temperature. The canonical partition function ZZ is defined as the sum over all microstates: Z(β)=∑i∈Se−βEiZ(\beta) = \sum_{i \in \mathcal{S}} e^{-\beta E_i}.\n\nThe probability PiP_i of the system occupying a specific microstate ii is given by the Boltzmann distribution:\nPi=e−βEiZ(β)P_i = \frac{e^{-\beta E_i}}{Z(\beta)}\n\nFurthermore, the expectation value of any observable O\mathcal{O} is calculated as:\n⟨O⟩=∑i∈SOiPi=1Z(β)∑i∈SOie−βEi\langle \mathcal{O} \rangle = \sum_{i \in \mathcal{S}} \mathcal{O}_i P_i = \frac{1}{Z(\beta)} \sum_{i \in \mathcal{S}} \mathcal{O}_i e^{-\beta E_i}.
Let S\mathcal{S} be a system undergoing a reaction ∑i=1NνiXi=0\sum_{i=1}^{N} \nu_i X_i = 0, where XiX_i are chemical species and νi\nu_i are stoichiometric coefficients. Define the grand canonical partition function Z(μ,T,V)\mathcal{Z}(\mu, T, V) such that the chemical potential of species ii is μi=−(∂∂Niln⁡Z)T,V,Nj≠i\mu_i = -\left(\frac{\partial}{\partial N_i} \ln \mathcal{Z}\right)_{T, V, N_{j\neq i}}. The equilibrium constant KK for the reaction is defined by the ratio of activities (or concentrations) at equilibrium, K=∏i=1Naiνi/νirefK = \prod_{i=1}^{N} a_i^{\nu_i/\nu_i^\text{ref}}, where aia_i is the activity of species ii. Furthermore, KK is rigorously related to the standard Gibbs free energy change ΔG∘\Delta G^\circ by the fundamental thermodynamic relation derived from the partition function: $$ \Delta G^\circ = -RT \ln K = -k_B T \ln \left(\frac{\mathcal{Z}_{products}}{\mathcal{Z}_{reactants}}\right) \quad \text{or equivalently,} \quad \Delta G^\circ = -RT \ln \left(\frac{\prod_{i=1}^{N} a_i^{\nu_i}}{\prod_{j=1}^{N} a_j^{\nu_j}}\right) \text{ for } \sum \nu_i X_i = 0 \text{.
Let H\mathcal{H} be the Hilbert space describing the system's quantum states, and let H^:H→R\hat{H}: \mathcal{H} \to \mathbb{R} be the Hamiltonian operator. Define the inverse temperature β=1/(kBT)\beta = 1/(k_B T). The canonical partition function, ZZ, is rigorously defined as the trace of the thermal density operator ρ^=e−βH^\hat{\rho} = e^{-\beta \hat{H}}: \n\nZ=Tr(e−βH^)Z = \text{Tr}(e^{-\beta \hat{H}}) \n\nIf the system is classical and the phase space is Γ\Gamma, the partition function is given by the integral over the phase space, weighted by the Boltzmann factor and the phase space volume element dΓd\Gamma: \n\nZ=1h3NN!∫Γe−βH(q,p)dqdpZ = \frac{1}{h^{3N} N!} \int_{\Gamma} e^{-\beta \mathcal{H}(\mathbf{q}, \mathbf{p})} d\mathbf{q} d\mathbf{p} \n\nwhere H(q,p)\mathcal{H}(\mathbf{q}, \mathbf{p}) is the classical Hamiltonian, and hh is Planck's constant, ensuring proper normalization for quantum-to-classical correspondence.
Let (Γ,B,μ)(\Gamma, \mathcal{B}, \mu) be a measure space representing the phase space, where μ\mu is the invariant measure associated with the Hamiltonian flow x(t)\mathbf{x}(t). Let A:Γ→RA: \Gamma \to \mathbb{R} be a continuous observable function. The Ergodic Hypothesis asserts that for almost every initial point x0∈Γ\mathbf{x}_0 \in \Gamma (with respect to the measure μ\mu), the time average of AA equals the phase space average of AA: \n\nlim⁡T→∞1T∫0TA(x(t;x0))dt=∫ΓA(x)dμ(x)\lim_{T \to \infty} \frac{1}{T} \int_0^T A(\mathbf{x}(t; \mathbf{x}_0)) dt = \int_{\Gamma} A(\mathbf{x}) d\mu(\mathbf{x})\n\nThis equality holds provided the flow x(t)\mathbf{x}(t) is ergodic with respect to the measure μ\mu, meaning that for any measurable set E⊂ΓE \subset \Gamma such that x(t)∈E\mathbf{x}(t) \in E for almost all tt, the measure μ(E)\mu(E) must be either 0 or μ(Γ)\mu(\Gamma).
Let ri∈R3\mathbf{r}_i \in \mathbb{R}^3 and \mathbf{p}_i \in \namespace{T}\mathbb{R}^3 be the position and momentum of the ii-th particle, respectively, for i=1,…,Ni=1, \dots, N. Define the Hamiltonian H(r,p)\mathcal{H}(\mathbf{r}, \mathbf{p}) for the ideal gas system as H=∑i=1Npi22m\mathcal{H} = \sum_{i=1}^{N} \frac{\mathbf{p}_i^2}{2m}. The canonical partition function is Z(β,V)=1N!h3N∫e−βHd3Npd3NrZ(\beta, V) = \frac{1}{N! h^{3N}} \int e^{-\beta \mathcal{H}} d^{3N}p d^{3N}r, where β=1/(kBT)\beta = 1/(k_B T). The probability density function P(v)P(\mathbf{v}) for the velocity vector v=(vx,vy,vz)\mathbf{v} = (v_x, v_y, v_z) of a single particle is given by the Maxwell-Boltzmann distribution:\n\nP(v)=(m2πkBT)3/2e−m(v)2/(2kBT)P(\mathbf{v}) = \left(\frac{m}{2\pi k_B T}\right)^{3/2} e^{-m(\mathbf{v})^2 / (2k_B T)}\n\nFurthermore, the expected value of the kinetic energy ⟨Ek⟩\langle E_k \rangle is derived from the equipartition theorem, yielding ⟨Ek⟩=32kBT\langle E_k \rangle = \frac{3}{2} k_B T.
Let S\mathcal{S} be a thermodynamic system characterized by state variables (T,P,N)(T, P, \mathbf{N}), where TT is temperature, PP is pressure, and N\mathbf{N} is the vector of particle numbers. The Gibbs Free Energy GG is defined as the thermodynamic potential such that its differential change dGdG is given by:\n\ndG=−SdT+VdP+∑iμidNidG = -S dT + V dP + \sum_{i} \mu_{i} dN_{i}\n\nwhere SS is the entropy, VV is the volume, and μi\mu_{i} is the chemical potential of species ii. Furthermore, GG is mathematically related to the internal energy UU via the Legendre transformation:\n\nG(T,P,N)=U(S,V,N)+TS−PVG(T, P, \mathbf{N}) = U(S, V, \mathbf{N}) + T S - P V\n\nFor an infinitesimal process, the criterion for spontaneity is ΔG≤0\Delta G \le 0, with equality holding at equilibrium, which implies that the maximum entropy principle dictates the system evolution along the path minimizing GG for fixed TT and PP.
Let x=(q,p)∈R2N\mathbf{x} = (\mathbf{q}, \mathbf{p}) \in \mathbb{R}^{2N} be the phase space coordinates, where q=(q1,…,qN)\mathbf{q} = (q_1, \dots, q_N) and p=(p1,…,pN)\mathbf{p} = (p_1, \dots, p_N). Assume the system evolves according to the Hamiltonian H(q,p)H(\mathbf{q}, \mathbf{p}). The flow x(t)\mathbf{x}(t) is generated by the Hamiltonian vector field v=(q˙,p˙)\mathbf{v} = (\dot{\mathbf{q}}, \dot{\mathbf{p}}), where q˙=∂H/∂p\dot{\mathbf{q}} = \partial H / \partial \mathbf{p} and p˙=−∂H/∂q\dot{\mathbf{p}} = -\partial H / \partial \mathbf{q}. The theorem asserts that the divergence of this vector field vanishes: ∇⋅v=∑i=1N(∂q˙i∂qi+∂p˙i∂pi)=0\nabla \cdot \mathbf{v} = \sum_{i=1}^{N} \left( \frac{\partial \dot{q}_i}{\partial q_i} + \frac{\partial \dot{p}_i}{\partial p_i} \right) = 0 Consequently, the phase space density ρ(x,t)\rho(\mathbf{x}, t) of an ensemble evolving under this flow satisfies the continuity equation: ∂ρ∂t+∑i=1N(∂∂qi(ρq˙i)+∂∂pi(ρp˙i))=0\frac{\partial \rho}{\partial t} + \sum_{i=1}^{N} \left( \frac{\partial}{\partial q_i} (\rho \dot{q}_i) + \frac{\partial}{\partial p_i} (\rho \dot{p}_i) \right) = 0 which implies the conservation of the phase space volume element Ω\Omega: ddt(∫Ωρ(x,t)d2Nx)=0\frac{d}{dt} \left( \int_{\Omega} \rho(\mathbf{x}, t) d^{2N}\mathbf{x} \right) = 0