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Kinetic Theory

Field: Statistical Mechanics

A theory that explains the physical properties of matter in terms of the motion of its constituent particles.

Sequence of Expressions

Let the system's configuration space be Q⊂R3N\mathcal{Q} \subset \mathbb{R}^{3N} and the corresponding phase space be Γ⊂R6N\Gamma \subset \mathbb{R}^{6N}. The generalized coordinates are defined by the vector q=(q1,q2,…,q3N)\mathbf{q} = (q_1, q_2, \dots, q_{3N}) and the conjugate momenta p=(p1,p2,…,p3N)\mathbf{p} = (p_1, p_2, \dots, p_{3N}). The number of degrees of freedom, ff, is the dimension of the configuration space, f=dim⁡(Q)f = \dim(\mathcal{Q}). For a system of NN particles interacting via a potential V(q)V(\mathbf{q}), the Hamiltonian is given by H(q,p)=∑i=1Npi22mi+V(q)\mathcal{H}(\mathbf{q}, \mathbf{p}) = \sum_{i=1}^{N} \frac{\mathbf{p}_i^2}{2m_i} + V(\mathbf{q}). The canonical volume element in phase space is dΓ=d3Nqd3Npd\Gamma = d^{3N}\mathbf{q} d^{3N}\mathbf{p}. The partition function ZZ is defined by the integral over the phase space: Z=1h3NN!∫Γe−βH(q,p)dΓZ = \frac{1}{h^{3N} N!} \int_{\Gamma} e^{-\beta \mathcal{H}(\mathbf{q}, \mathbf{p})} d\Gamma where β=1/kBT\beta = 1/k_B T. The number of degrees of freedom ff dictates the exponent of the momentum integral, such that the classical limit of the partition function yields Z∝(1ℏβ)f/2⋅constantZ \propto (\frac{1}{\hbar \beta})^{f/2} \cdot \text{constant}.
Let v=(vx,vy,vz)∈R3\textbf{v} = (v_x, v_y, v_z) \in \mathbb{R}^3 be the velocity vector of a particle of mass mm in a gas at temperature TT. The probability density function f(v)f(\textbf{v}) for this velocity, derived from the canonical ensemble, is given by:\\begin{equation} f(\textbf{v}) = \left(\frac{m}{2\pi k_B T}\right)^{\frac{3}{2}} e^{-\frac{m(v_x^2 + v_y^2 + v_z^2)}{2k_B T}} \end{equation}\.\\text{The normalization condition requires that the integral over all velocity space equals unity:}\ \nobreak \int_{\mathbb{R}^3} f(\textbf{v}) \, d^3\textbf{v} = 1.\\text{The distribution of the speed magnitude } v = |\textbf{v}| \text{ is given by the radial probability density function } P(v) = 4\pi v^2 f(\textbf{v}) \text{, which yields:}\ \nobreak P(v) = \left(\frac{m}{2\pi k_B T}\right)^{\frac{3}{2}} 4\pi v^2 e^{-\frac{m v^2}{2k_B T}} \text{ for } v \ge 0.
Let H(p,q)\mathcal{H}(\mathbf{p}, \mathbf{q}) be the Hamiltonian of a system of NN particles in a volume VV, where p\mathbf{p} and q\mathbf{q} are the canonical momentum and position vectors, respectively. The average kinetic energy ⟨K⟩\langle K \rangle is defined by the expectation value over the phase space Γ\Gamma: ⟨K⟩=1Z∫e−βH(p,q)(∑i=1Npi22m)dΓ\langle K \rangle = \frac{1}{Z} \int e^{-\beta \mathcal{H}(\mathbf{p}, \mathbf{q})} \left( \sum_{i=1}^{N} \frac{\mathbf{p}_i^2}{2m} \right) d\Gamma. For a system where the potential energy U(q)U(\mathbf{q}) is independent of momentum, the partition function ZZ is separable, and the average kinetic energy is given by ⟨K⟩=32NkBT\langle K \rangle = \frac{3}{2} N k_B T. Alternatively, using the equipartition theorem, for each quadratic degree of freedom xix_i, the average energy is ⟨12kBT∂2H∂xi2⟩=12kBT\langle \frac{1}{2} k_B T \frac{\partial^2 \mathcal{H}}{\partial x_i^2} \rangle = \frac{1}{2} k_B T. Thus, the total average kinetic energy is ⟨K⟩=12kBT∑i=13N∂2H∂xi21kBT⋅32NkBT\langle K \rangle = \frac{1}{2} k_B T \sum_{i=1}^{3N} \frac{\partial^2 \mathcal{H}}{\partial x_i^2} \frac{1}{k_B T} \cdot \frac{3}{2} N k_B T.
Let H:Phase Space→R\mathcal{H}: \text{Phase Space} \to \mathbb{R} be the Hamiltonian operator for a system of NN particles. Consider the canonical ensemble defined by the partition function Z(β)=Tr(e−βH)Z(\beta) = \text{Tr}\left(e^{-\beta \mathcal{H}}\right), where β=1/(kBT)\beta = 1/(k_B T). The average internal energy ⟨E⟩\langle E \rangle is given by ⟨E⟩=−∂∂βlnZ(β)\langle E \rangle = -\frac{\partial}{\partial \beta} \text{ln} Z(\beta). By definition, the thermodynamic temperature TT is related to the energy and entropy SS via 1T=(∂S∂⟨E⟩)V,N\frac{1}{T} = \left(\frac{\partial S}{\partial \langle E \rangle}\right)_{V, N}. Since S=kBlnZ+β⟨E⟩S = k_B \text{ln} Z + \beta \langle E \rangle, we have 1T=∂∂⟨E⟩(kBlnZ+β⟨E⟩)\frac{1}{T} = \frac{\partial}{\partial \langle E \rangle} \left(k_B \text{ln} Z + \beta \langle E \rangle\right). Equating the two expressions for 1/T1/T yields the definition of the Boltzmann constant kBk_B as:\n\nkB=⟨E⟩T(∂lnZ(β)∂⟨E⟩)β or equivalently, kB=⟨E⟩T(∂β∂⟨E⟩)βk_B = \frac{\langle E \rangle}{T} \left( \frac{\partial \text{ln} Z(\beta)}{\partial \langle E \rangle} \right)_{\beta} \text{ or equivalently, } k_B = \frac{\langle E \rangle}{T} \left( \frac{\partial \beta}{\partial \langle E \rangle} \right)_{\beta}
Let nn be the number density of particles in a volume VV, and let dd be the effective molecular diameter. Define the collision cross-section σ\sigma as σ=πd2\sigma = \pi d^2. The collision frequency ν\nu (collisions per unit time per particle) is given by the integral over relative velocities v⃗rel\vec{v}_{rel}: ν=⟨v⃗rel⋅A⃗⟩/⟨r⃗⋅A⃗⟩\nu = \langle \vec{v}_{rel} \cdot \vec{A} \rangle / \langle \vec{r} \cdot \vec{A} \rangle, where A⃗\vec{A} is the relative velocity vector and ⟨… ⟩\langle \dots \rangle denotes the ensemble average. For an ideal gas, the mean free path λ\lambda is defined as the average particle speed vˉ\bar{v} divided by the collision frequency ν\nu. Specifically, λ\lambda is derived from the relationship:\n\nλ=12πd2n\lambda = \frac{1}{\sqrt{2} \pi d^2 n}
Let H(θ)\mathcal{H}(\boldsymbol{\theta}) be the Hamiltonian of a system with generalized coordinates θ=(θ1,…,θN)\boldsymbol{\theta} = (\theta_1, \dots, \theta_N) in thermal equilibrium at temperature TT. Assume the contribution of the ii-th degree of freedom to the Hamiltonian is quadratic, Hi=12kiθi2H_i = \frac{1}{2} k_i \theta_i^2. The canonical partition function is Z=∫e−H(θ)/kBTdθZ = \int e^{-\mathcal{H}(\boldsymbol{\theta})}/k_B T \text{d}\boldsymbol{\theta}. The average energy ⟨Ei⟩\langle E_i \rangle associated with this degree of freedom is given by the expectation value: ⟨Ei⟩=1Z∫e−H(θ)/kBT(∂H∂θiθi)dθ\langle E_i \rangle = \frac{1}{Z} \int e^{-\mathcal{H}(\boldsymbol{\theta})}/k_B T \left( \frac{\partial \mathcal{H}}{\partial \theta_i} \theta_i \right) \text{d}\boldsymbol{\theta}. Under the assumption that the system is ergodic and the Hamiltonian is separable into quadratic terms, the theorem states that for every such degree of freedom ii: ⟨Ei⟩=12kBT \langle E_i \rangle = \frac{1}{2} k_B T
Let S\mathcal{S} be a system of NN non-interacting particles of mass mm confined to a volume VV at temperature TT. The system's Hamiltonian is H=∑i=1Np⃗i22m\mathcal{H} = \sum_{i=1}^{N} \frac{\vec{p}_i^2}{2m}. The pressure PP is defined by the average momentum flux tensor ⟨Π⟩\langle \Pi \rangle exerted on the container walls. For an ideal gas, the equation of state is derived from the partition function ZZ: Z=1N!(Vh3)N(2πmkBTh2)3N/2Z = \frac{1}{N!}\left(\frac{V}{h^3}\right)^N \left(\frac{2\pi m k_B T}{h^2}\right)^{3N/2} where kBk_B is the Boltzmann constant and hh is Planck's constant. The internal energy UU is related to ZZ by U=−(∂ln⁡Z∂β)V,NU = -\left(\frac{\partial \ln Z}{\partial \beta}\right)_{V, N}, where β=1/(kBT)\beta = 1/(k_B T). By the equipartition theorem and the ideal gas law, the pressure PP is given by the thermodynamic relation: P=NkBTV=nRTVP = \frac{N k_B T}{V} = \frac{n R T}{V} where n=N/NAn = N/N_A is the number density and R=NAkBR = N_A k_B is the universal gas constant.
Let S\mathcal{S} be a system of NN particles of mass mm confined to a volume VV at temperature TT. The phase space is Γ={(q,p)∈R3N×R3N}\Gamma = \{({\mathbf{q}}, \mathbf{p}) \in \mathbb{R}^{3N} \times \mathbb{R}^{3N}\}. The Hamiltonian is H(q,p)=∑i=1Npi22m+U(q)\mathcal{H}({\mathbf{q}}, \mathbf{p}) = \sum_{i=1}^{N} \frac{\mathbf{p}_i^2}{2m} + U(\mathbf{q}). Assuming the system is in thermal equilibrium, the probability density function is the canonical ensemble distribution ρ∝e−H/kBT\rho \propto e^{-\mathcal{H}/k_B T}. The average kinetic energy ⟨K⟩\langle K \rangle is calculated as ⟨K⟩=∫Γ(∑i=1Npi22m)ρdqdp/Z\langle K \rangle = \int_{\Gamma} \left(\sum_{i=1}^{N} \frac{\mathbf{p}_i^2}{2m}\right) \rho d\mathbf{q} d\mathbf{p} / Z. By the equipartition theorem, ⟨K⟩=32NkBT\langle K \rangle = \frac{3}{2} N k_B T. For a single particle, the average kinetic energy is ⟨K1⟩=12m⟨v2⟩=32kBT\langle K_1 \rangle = \frac{1}{2} m \langle v^2 \rangle = \frac{3}{2} k_B T. The Root-Mean-Square speed, vrmsv_{rms}, is defined by the expectation value of the squared speed: vrms=⟨v2⟩v_{rms} = \sqrt{\langle v^2 \rangle}. Therefore, vrms=3kBTmv_{rms} = \sqrt{\frac{3 k_B T}{m}}.
Let H:R3N→R\mathcal{H}: \mathbb{R}^{3N} \to \mathbb{R} be the Hamiltonian of a system of NN particles. Define the inverse temperature β=1/(kBT)\beta = 1/(k_B T). The canonical ensemble probability density function ρ(q,p)\rho(\boldsymbol{q}, \boldsymbol{p}) for the phase space coordinates (q,p)∈R3N(\boldsymbol{q}, \boldsymbol{p}) \in \mathbb{R}^{3N} is given by: ρ(q,p)=e−βH(q,p)Z(β)\rho(\boldsymbol{q}, \boldsymbol{p}) = \frac{e^{-\beta \mathcal{H}(\boldsymbol{q}, \boldsymbol{p})}}{Z(\beta)}\nwhere the canonical partition function Z(β)Z(\beta) is the normalization constant: Z(β)=∫ ⁣ ⁣e−βH(q,p) d3Nqd3NpZ(\beta) = \int \!\! e^{-\beta \mathcal{H}(\boldsymbol{q}, \boldsymbol{p})} \, d^{3N} \boldsymbol{q} d^{3N} \boldsymbol{p}\nFor any observable O(q,p)\mathcal{O}(\boldsymbol{q}, \boldsymbol{p}), its ensemble average ⟨O⟩\langle \mathcal{O} \rangle is calculated as: ⟨O⟩=∫ ⁣ ⁣O(q,p)ρ(q,p) d3Nqd3Np\langle \mathcal{O} \rangle = \int \!\! \mathcal{O}(\boldsymbol{q}, \boldsymbol{p}) \rho(\boldsymbol{q}, \boldsymbol{p}) \, d^{3N} \boldsymbol{q} d^{3N} \boldsymbol{p}\nFurthermore, the Helmholtz free energy AA is related to Z(β)Z(\beta) by: A(β)=−kBTln⁡Z(β)A(\beta) = -k_B T \ln Z(\beta)\nThis framework establishes the link between the microscopic dynamics (via H\mathcal{H}) and the macroscopic thermodynamic potential AA.
Let f(r,v,t)f(\mathbf{r}, \mathbf{v}, t) be the single-particle distribution function in phase space Γ=R3×R3\Gamma = \mathbb{R}^3 \times \mathbb{R}^3. The evolution of ff is governed by the Boltzmann equation: ∂f∂t+v⋅∇rf+Fm⋅∇vf=(∂f∂t)coll \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{r}} f + \frac{\mathbf{F}}{m} \cdot \nabla_{\mathbf{v}} f = \left(\frac{\partial f}{\partial t}\right)_{coll} where F\mathbf{F} is the external force. The collision integral, (∂f∂t)coll\left(\frac{\partial f}{\partial t}\right)_{coll}, quantifies the rate of change due to particle interactions and is defined as: (∂f∂t)coll=∫v′∫Ωσ(g,g′,v,v′)[f′f′−ff′]⋅g⋅dgdv′ \left(\frac{\partial f}{\partial t}\right)_{coll} = \int_{\mathbf{v}'} \int_{\Omega} \sigma(\mathbf{g}, \mathbf{g}', \mathbf{v}, \mathbf{v}') \left[ f' f' - f f' \right] \cdot \mathbf{g} \cdot d\mathbf{g} d\mathbf{v}' Here, g=v−v′\mathbf{g} = \mathbf{v} - \mathbf{v}' is the relative velocity, σ(g,g′,v,v′)\sigma(\mathbf{g}, \mathbf{g}', \mathbf{v}, \mathbf{v}') is the differential cross-section for collision between particles with relative velocities g\mathbf{g} and g′\mathbf{g}', and dgdv′d\mathbf{g} d\mathbf{v}' represents the integration over all possible outgoing velocities and incoming velocities, respectively. The theory postulates that the collision frequency ν\nu is proportional to the integral of the cross-section over the relative velocity distribution.