Statistical physics

A superfluid exhibits superfluidity, with phase stiffness and a phase-gradient current. In an appropriate ordered or quasi-ordered regime it supports low-speed flow without viscous dissipation, and its long-wavelength Goldstone boson has a sound-like dispersion. True thermodynamic phase selection and Bose-Einstein condensation need separate infrared conditions.

Superfluid velocity ( $u = (ℏ/ m) \nabla S$ )

For a condensate

ψ = n e^{i S}

with kinetic operator

- ℏ^{2} \nabla^{2} / (2 m)

, its particle current is

n u

with the displayed velocity. This follows by subtracting the Gross–Pitaevskii equation from its complex conjugate. More generally, in units where the kinetic operator is

- D \nabla^{2}

, the current velocity is

2 D \nabla S

. The complex argument gradient and current velocity coincide only when

D = 1/2

Superfluid order-parameter effective theory

For a complex superfluid order parameter

ψ = n e^{i θ}

, the nonrelativistic Landau-Ginzburg kinetic term contains

- n \dot{θ}

. Thus number density and phase are canonically conjugate, so a phase-selected state necessarily fluctuates in particle number.

Number-phase conjugacy

Number-phase conjugacy is the canonical relation between a conserved

U (1)

charge and the phase of its order parameter. A state of sharp number cannot also have a sharply selected broken-symmetry phase.

Thermal phase fluctuation

A thermal phase fluctuation is a finite-temperature fluctuation of the phase of a complex order parameter. For a linearly dispersing phase mode, its infrared variance is proportional to

T \int d^{d} k / k^{2}

, which prevents true finite-temperature long-range order when

d \leq 2

Superconductivity

 2  0

Superconductivity is a phase with vanishing electrical resistance and expulsion of magnetic flux from the bulk, the Meissner effect. Conventional microscopic BCS theory describes pairing of electrons into Cooper pairs.

Kinetic theory ( $f (t, x, v)$ )

Describe many-particle systems through distributions on phase space. Transport moves particles along trajectories, while collision or diffusion terms modify their velocity distributions. The Boltzmann equation, Hamiltonian Liouville equation and Fokker-Planck equation are central models.

Boltzmann-Grad limit ( $N a^{d - 1} \sim 1, N a^{d} \to 0$ )

For

N

hard particles in a fixed macroscopic volume, shrink their diameter

a

while keeping

N a^{d - 1}

of order one. The occupied volume fraction tends to zero, but the mean free path stays macroscopic. Deriving the Boltzmann equation in this dilute limit requires control of recollisions and propagation of an incoming independence hypothesis related to molecular chaos for the Boltzmann equation. Its validity is not an automatic all-time consequence of small particle size.

Knudsen number ( $Kn = ℓ_{mfp} / L$ )

The ratio of the mean free path to a macroscopic length compares collision and transport scales. Small Knudsen number motivates the hydrodynamic limit of the Boltzmann equation; large values favor a direct kinetic description. The scaling of time, velocity and space must also be stated when using it as a coefficient in a kinetic equation.

Molecular chaos for the Boltzmann equation

The incoming two-particle distribution at a collision is approximated by the product of one-particle distributions. This closes the binary collision term into the quadratic nonlinear Boltzmann collision operator. It does not assert exact independence after collisions: outgoing pairs can be correlated, and time reversal generally converts those correlations into incoming correlations. A rigorous kinetic limit needs propagation and convergence estimates rather than a bare independence assertion.

Unordered-pair collision rate ( $Γ_{ii} = K_{ii} N_{i} (N_{i} - 1) /2$ )

For a dilute phase-mixed population, an identical-species collision rate counts unordered pairs:

Γ_{ii} = K_{ii} N_{i} (N_{i} - 1) /2

. Different species have rate

Γ_{ij} = K_{ij} N_{i} N_{j}

. Here

K_{ij} = ⟨ σ_{ij} v_{ij} ⟩ / V

uses the collision cross-section and relative speed. Destroying both identical particles gives

\dot{N}_{i} = - 2 Γ_{ii}

, not

- Γ_{ii}

. This distinction prevents inconsistent factors when using one kinetic kernel for several particle sizes.

Kac model

This many-particle stochastic collision model rotates one uniformly selected velocity pair by a uniformly selected angle, conserving the sum of squared velocities. Its normalized angular averaging is the Kac collision operator, and its continuous-time evolution is the Kac master equation. In this version velocities range over

R^{N}

and initial energy is not restricted to one sphere; the evolution preserves the initial energy distribution.

Relative entropy in Kac's model ( $H_{N} (F) = \int F lo g (F / γ_{N})$ )

Use

γ_{N} = (2 π)^{- N /2} e^{- ∣ v ∣^{2} /2}

as the reference probability density. The relative entropy is nonnegative by writing

F = γ_{N} u

and integrating

u lo g u - u + 1

. Its expansion is

\int F lo g F + (N /2) lo g (2 π) + (1/2) \int ∣ v ∣^{2} F

. Equality holds only at

F = γ_{N}

, whereas other radial densities may still be stationary for the Kac master equation.

Kac entropy production ( $D_{N} (F) = - \frac{d}{d t} H_{N} (F) \geq 0$ )

For the Kac master equation and finite differentiable entropy, mass and energy conservation remove the non-logarithmic entropy terms. Symmetrizing each pair rotation then gives

D_{N} (F) = [2 π (N - 1)]^{- 1} \sum_{i < j} \int\int (F \circ R_{ij, θ} - F) lo g ((F \circ R_{ij, θ}) / F) d v d θ

. The integrand is nonnegative by monotonicity of the logarithm, with the usual extended convention at zeros. Zero dissipation corresponds to radiality; it need not imply zero relative entropy.

Kac chaos ( $Π_{k} (F_{N}) ⇀ f^{\otimes k}$ )

A permutation-symmetric sequence of probability laws is chaotic with one-particle law

f

when, for every fixed

k

, its

k

-coordinate marginal distributions converge weakly to the product law

f^{\otimes k}

N \to \infty

. This is an asymptotic independence condition, not a consequence of symmetry alone. A rigorous kinetic limit also requires propagation of this property under the dynamics and sufficient convergence to pass collision integrals.

Kac master equation ( $\partial_{t} F_{N} = N (Q - I) F_{N}$ )

The jump process with total collision rate

N

and normalized pair-angle operator

Q

evolves by

\partial_{t} F_{N} = N (Q - I) F_{N}

. It preserves mass and the total squared velocity. Its one-coordinate distribution satisfies the Kac marginal evolution equation, and its Gaussian relative entropy in Kac's model decreases at the rate given by Kac entropy production.

Kac marginal evolution equation ( $\partial_{t} Π_{1} = π^{- 1} \int (Π_{2} \circ R_{θ} - Π_{2}) d v_{2} d θ$ )

For a permutation-symmetric density in the Kac master equation, integrate over all but the first velocity. Collisions among integrated coordinates cancel. The

N - 1

pairs containing the first coordinate give equal terms, with coefficient

N (N - 1) / (2 π (2 N)) = 1/ π

. The resulting exact evolution depends on the two-coordinate marginal distribution; it is not a closed equation for the first marginal without additional asymptotic independence.

Kac collision operator ( $Q = (2 N)^{- 1} \sum_{i < j} (2 π)^{- 1} \int U_{ij, θ} d θ$ )

For

N \geq 2

U_{ij, θ} F = F \circ R_{ij, θ}

uses the genuine two-coordinate rotation

(v_{i}, v_{j}) \mapsto (v_{i} cos θ + v_{j} sin θ, - v_{i} sin θ + v_{j} cos θ)

. Average it uniformly over angles and pairs to obtain

Q

. Rotation invariance of Lebesgue measure makes

Q

a bounded self-adjoint operator of norm one on

L^{2} (R^{N})

. Each pair average is an orthogonal projection, but their full average is not in general a projection.

Dirichlet form of the Kac collision operator ( $⟨ F, (I - Q) F ⟩ \geq 0$ )

For the Kac collision operator, expanding the squared Hilbert space norm gives

⟨ F, (I - Q) F ⟩ = (4 π (2 N))^{- 1} \sum_{i < j} \int_{0}^{2 π} \int_{R^{N}} ∣ F (R_{ij, θ} v) - F (v) ∣^{2} d v d θ

. The nonnegative form measures failure of invariance under the pair rotations. It vanishes precisely on the radial kernel of the Kac collision operator.

Radial kernel of the Kac collision operator ( $ker (I - Q) = L_{radial}^{2} (R^{N})$ )

Zero Dirichlet form of the Kac collision operator forces invariance under every pair rotation; strong continuity upgrades almost every angle to every angle. Coordinate-plane rotations generate the special orthogonal group, whose action is transitive on each sphere for

N \geq 2

. Averaging over normalized Haar measure therefore identifies the invariant

L^{2}

functions with radial functions. Conversely radial functions are fixed by every pair rotation.

Free-transport phase mixing ( $f_{t} (k, ξ) = f_{0} (k, ξ + k t)$ )

On the unit-period spatial circle, free streaming has solution

f_{t} (x, v) = f_{0} (x - t v, v)

. With Fourier phase

e^{- 2 πi (k x + ξ v)}

, its mixed Fourier transform is

f_{t} (k, ξ) = f_{0} (k, ξ + k t)

. A nonzero spatial mode of the velocity-integrated density therefore samples increasingly large velocity frequencies. The conserved spatial zero mode remains, while extra velocity regularity supplies quantitative decay of the other modes.

Uniform phase-mixing bound from velocity derivatives ( $∥ ρ_{t} - ρ_{\infty} ∥_{\infty} \leq 2 ζ (m) ∥ \partial_{v}^{m} f_{0} ∥_{1} / (2 π ∣ t ∣)^{m}$ )

For

m \geq 2

integrable weak velocity derivatives on a unit-period spatial circle, the Fourier transform of a derivative gives

∣ f_{0} (k, k t) ∣ \leq ∥ \partial_{v}^{m} f_{0} ∥_{1} / (2 π ∣ k t ∣)^{m}

for

k \neq = 0

. Summing these modes uses

\sum_{k \neq = 0} ∣ k ∣^{- m} = 2 ζ (m)

. The resulting absolute Fourier summability gives the displayed uniform bound for the continuous representative of the density. For

m = 1

the same summation diverges, so this argument needs additional input.

Mean velocity of a kinetic distribution ( $u = \frac{\int v f ( x , v ) d x d v}{\int f ( x , v ) d x d v}$ )

The mass-normalized first velocity moment of a distribution on phase space. For a velocity-only distribution, omit the position integral. Nonzero finite mass and a finite first moment are required. For a probability distribution this is its expected value; its unnormalized counterpart is the momentum.

Hamiltonian Liouville equation ( $\partial_{t} f + \nabla_{v} H \cdot \nabla_{x} f - \nabla_{x} H \cdot \nabla_{v} f = h$ )

Transport a phase-space density along Hamilton's equations. The Hamiltonian vector field is

b = (\nabla_{v} H, - \nabla_{x} H)

and the equation is

\partial_{t} f + b \cdot \nabla_{x, v} f = h

. Its zero divergence gives volume-preserving transport. For

h = 0

, the density is constant along characteristics; a source is integrated by the Duhamel formula for Hamiltonian transport. This is distinct from the elliptic Liouville equation.

Hyperbolic characteristic flow for an inverted oscillator ( $A_{t} = (cosh t sinh t sinh t cosh t)$ )

The Hamiltonian

(v^{2} - x^{2}) /2

gives

\overset{x}{˙} = v

\overset{v}{˙} = x

. Its characteristic flow map is multiplication by

A_{t}

, with

A_{t} A_{s} = A_{t + s}

and determinant one. The backward map is

A_{- t}

. This expanding-contracting linear flow preserves phase-space Lebesgue measure and every finite-

p

integral of a transported density, despite stretching its level sets.

Invariant-source secular growth in Hamiltonian transport ( $f_{t} = t h$ )

f_{0} = 0

and a nonzero time-independent source is invariant along the Hamiltonian flow, then the Duhamel formula for Hamiltonian transport gives

f_{t} = t h

. For the nonzero-frequency oscillator,

h = e^{- H_{ω}}

is smooth and belongs to all finite Lp spaces, yet its resulting solution has infinite space-time norm on

(0, \infty)

Finite-time Lp bound for Hamiltonian transport ( $∥ f (t) ∥_{p} \leq ∥ f_{0} ∥_{p} + \int_{0}^{t} ∥ h (s) ∥_{p} d s$ )

The Duhamel formula for Hamiltonian transport, phase space preservation and Minkowski integral inequality give the displayed bound. A time-independent source in an Lp space yields finite space-time norm on every bounded interval; it need not yield integrability on infinite time.

Spatially nonintegrable forcing in transport ( $h = x^{2} + v^{2}$ )

Smooth forcing need not be integrable in phase space. For the hyperbolic characteristic flow for an inverted oscillator, the source

x^{2} + v^{2}

accumulates as

q_{t} = \frac{1}{2} sinh (2 t) (x^{2} + v^{2}) - (cosh (2 t) - 1) xv

. Its smallest quadratic-form eigenvalue is

(1 - e^{- 2 t}) /2 > 0

for

t > 0

. Starting with a nonnegative integrable Gaussian function, the solution is therefore unbounded and outside every finite Lp space at positive times. Locally time-integrable

L^{p}

forcing prevents this failure by the Minkowski integral inequality.

Duhamel formula for Hamiltonian transport ( $f (t, z) = f_{0} (Φ (0, t, z)) + \int_{0}^{t} h (s, Φ (s, t, z)) d s$ )

For a complete invertible Hamiltonian flow

Φ (t, s)

, integrate the source along the backward characteristic ending at

z

at time

t

. The displayed formula solves the Hamiltonian Liouville equation. Volume preservation makes each composition with

Φ (s, t)

an isometry of Lp spaces.

Statistical Hamiltonian ( $P (ω) \propto e^{- β_{T} H (ω)}$ )

A statistical Hamiltonian assigns an energy to a microscopic configuration. Its Boltzmann weight defines the canonical ensemble, and summing or integrating that weight gives a partition function. A reduced Hamiltonian absorbs inverse temperature into its coefficients, so the weight is

e^{- H_{reduced}}

. An Ising model uses discrete Ising spin configurations, while Landau-Ginzburg theory uses a coarse scalar field.

Coarse-grained variable

A coarse-grained variable retains selected macroscopic information while averaging over unresolved microscopic states. At thermal equilibrium, the statistical multiplicity of those states contributes entropy to the effective Helmholtz free energy of the retained variable.

Statistical field theory

Thermodynamic limit

The thermodynamic limit sends particle number and volume to infinity at fixed intensive quantities such as number density. Distinct symmetry-breaking phases become sharply defined in this limit.

Virial expansion

The virial expansion expresses the free energy or pressure of a dilute system as a power series in number density whose coefficients encode few-body interactions.

Second virial coefficient ( $B_{2}$ )

For particles with pair potential

U

, the second virial coefficient is

B_{2} = \frac{1}{2} \int [1 - e^{- U (r) / (k_{B} T)}] d r,

with the integration measure adapted to the available relative configurations. Repulsion contributes positively and attraction negatively.

Cluster expansion

The cluster expansion organizes the logarithm of a dilute interacting system's canonical partition function as a sum over connected clusters built from Mayer bonds.

Pair potential ( $u (r)$ )

A pair potential assigns an interaction energy

u (∣ x_{i} - x_{j} ∣)

to each pair of particles. A pairwise-additive system has

U = \sum_{i < j} u (∣ x_{i} - x_{j} ∣)

Mayer f-function ( $f (r) = e^{- u (r) / (k_{B} T)} - 1$ )

The Mayer f-function packages the departure of a pair's Boltzmann factor from one and is the basic bond in the dilute-gas cluster expansion.

Hard-sphere gas

A hard-sphere gas has

u (r) = + \infty

below the contact separation and

u (r) = 0

above it, so particles cannot overlap.

Excluded volume

The excluded volume is the volume of relative positions forbidden to another particle. For hard spheres of radius

r_{0}

, that forbidden region is a sphere of radius

2 r_{0}

Free-energy density ( $f$ )

A free-energy density is free energy per unit volume. Its minima with respect to unconstrained order parameters determine homogeneous equilibrium states.

Spin model

A spin model assigns a discrete or continuous spin variable to each site of a lattice or vertex of a graph and gives interacting configurations a Boltzmann factor. Spin models describe collective order and phase transitions in magnetic and other many-body systems.

Blume–Emery–Griffiths model

The classical Blume–Emery–Griffiths model uses spins

σ_{i} \in {- 1, 0, 1}

with both bilinear

σ_{i} σ_{j}

and biquadratic

σ_{i}^{2} σ_{j}^{2}

nearest-neighbour interactions, together with a single-site term proportional to

σ_{i}^{2}

. One convention is

H = - J \sum σ_{i} σ_{j} - L \sum σ_{i}^{2} σ_{j}^{2} - D \sum σ_{i}^{2} - h \sum σ_{i}

. Signs of the single-site parameter vary across conventions. Setting

L = 0

gives a Blume–Capel model after matching that sign convention. The one-dimensional version admits an exact spin-chain transfer matrix.

Transfer matrix for a classical spin chain ( $Z_{N} = tr W^{N}$ )

For a nearest-neighbour classical chain with finitely many spin states and dimensionless bond energy

E (s, t)

, its transfer matrix is

W_{s t} = e^{- E (s, t)}

. On-site energies are divided between adjacent bonds. Periodic boundary conditions give partition function

Z_{N} = tr W^{N}

by direct multiplication and summation over spin labels. For a strictly positive symmetric matrix, the Perron–Frobenius theorem gives

lim_{N \to \infty} N^{- 1} lo g Z_{N} = lo g λ_{m a x}

. A diagonal local observable

O

has expectation

tr (O W^{N}) / Z_{N}

, tending to

v_{m a x}^{T} O v_{m a x}

for a normalized dominant eigenvector.

Spin magnetization ( $M = n^{- 1} \sum_{i = 1}^{n} S_{i}$ )

The spin magnetization is the average signed spin of a discrete spin system. For

S_{i} \in {- 1, 1}

k

positive spins give

M = - 1 + 2 k / n

. Multiplying by the magnetic moment per spin and the spin density relates this dimensionless average to physical magnetization.

Spin inversion symmetry

Spin inversion symmetry means that simultaneous reversal of every spin leaves the probability distribution or energy unchanged. For a finite spin model with this symmetry, the weights of opposite configurations pair up, so every observable that changes sign under the reversal has zero expected value. An applied magnetic field generally breaks this symmetry.

Ising model

 2  0

The Ising model assigns spins

σ_{v} \in {- 1, 1}

to graph vertices with probability proportional to

exp (β \sum_{uv} σ_{u} σ_{v})

in zero external field.

Ising spin variable ( $σ_{i} \in {- 1, 1}$ )

An Ising Ising spin variable is a classical two-state degree of freedom. Its thermal mean can lie anywhere in

[- 1, 1]

even though each microscopic value is

- 1

1

. It should not be confused with an arbitrary quantum-spin operator. Products of neighbouring Ising spins supply the pair interactions of the Ising model, and their average defines spin magnetization.

Mean-field theory of the Ising model ( $m = tanh [β_{T} (h + q J m)]$ )

The mean-field approximation neglects products of Ising spin deviations, replacing each Ising spin's neighbours by the common spin magnetization. With coordination number of a lattice

q

, the effective field is

h + q J m

. The self-consistency equation equates the assumed mean Ising spin with its independent-spin expectation in that effective field. It predicts

k_{B} T_{c} = q J

and the usual quartic mean-field critical exponents, while neglecting spatially correlated fluctuations.

Bragg-Williams free energy of the Ising model ( $A_{BW} = ⟨ H ⟩ - TS$ )

A product trial distribution with actual mean Ising spin

m

gives

\frac{A _{BW}}{N} = - \frac{q J}{2} m^{2} - hm + k_{B} T [\frac{1 + m}{2} lo g \frac{1 + m}{2} + \frac{1 - m}{2} lo g \frac{1 - m}{2}] .

Its entropy is the sum of independent-spin entropies, with

0 lo g 0 = 0

. Differentiation gives

k_{B} T arctanh m = q J m + h

, the self-consistency equation. The variational mean parameter is distinct from the assumed field parameter of the Ising auxiliary mean-field free energy until the equilibrium condition is imposed.

Ising auxiliary mean-field free energy ( $A_{aux} / N = q J m^{2} /2 - k_{B} T lo g [2 cosh (β_{T} (h + q J m))]$ )

The approximate partition function obtained by decoupling each Ising bond has this free energy. The quadratic constant compensates double counting. Its stationary condition is the self-consistency equation; away from stationarity

m

parametrizes the assumed effective field and need not equal that distribution's mean Ising spin. It agrees at stationary points with the Bragg-Williams free energy of the Ising model, although their off-equilibrium expansions differ.

Curie–Weiss model ( $P (s) = Z_{n, β}^{- 1} e^{β (\sum_{i} s_{i})^{2} / (2 n)}$ )

The zero-field ferromagnetic Curie–Weiss model is an Ising model on

n

spins with an equal interaction between every pair. For

s_{i} \in {- 1, 1}

and

β > 0

, its normalized probability mass function is

Z_{n, β}^{- 1} exp [β (\sum_{i} s_{i})^{2} / (2 n)]

. The partition function

Z_{n, β}

sums these weights over all configurations. The square includes constant diagonal terms; these affect only normalization. Spin inversion symmetry makes every spin's expected value zero, while grouping configurations by their spin magnetization gives

P (M = m) = Z_{n, β}^{- 1} (n ( 1 + m ) /2 n) e^{β n m^{2} /2} .

Here

m = - 1 + 2 k / n

0 \leq k \leq n

Transfer matrix for the one-dimensional Ising model

For nearest-neighbour interactions, a two-by-two transfer matrix assigns one Boltzmann factor to every adjacent spin pair. With periodic boundary conditions, summing all spin configurations gives the partition function as the trace of a power of this matrix.

Blume–Capel model

The Blume–Capel model is a spin model with

σ_{i} \in {- 1, 0, 1}

and Hamiltonian

H = - J \sum_{⟨ i, j ⟩} σ_{i} σ_{j} + D \sum_{i} σ_{i}^{2} - h \sum_{i} σ_{i} .

Competition between the Ising interaction and the single-ion anisotropy

D

produces a tricritical point in its phase diagram.

Blume–Capel mean-field tricritical parameters ( $T_{TCP} = z J / (3 k_{B})$ )

For ferromagnetic exchange

J

and coordination

z

, the mean-field approximation has

f = z J M^{2} /2 - k_{B} T lo g [1 + 2 e^{- g / (k_{B} T)} cosh (z J M / (k_{B} T))]

. With

K = z J / (k_{B} T)

and

q = 2 e^{- g / (k_{B} T)} / [1 + 2 e^{- g / (k_{B} T)}]

, its quadratic and quartic coefficients vanish at

q = 1/3

K = 3

. The sextic coefficient of

f / (k_{B} T)

9/40 > 0

. Hence

T_{TCP} = z J / (3 k_{B})

and

g_{TCP} = k_{B} T_{TCP} lo g 4

; these are not exact finite-dimensional lattice values.

Glauber dynamics

Glauber dynamics is a local Markov chain for a spin system. In its heat-bath form, each step chooses one vertex and resamples that spin from its conditional Gibbs distribution given all other spins.

Free boundary condition

A free boundary condition retains only interactions whose endpoints both lie in the finite volume and imposes no exterior spin values.

Plus boundary condition

A plus boundary condition fixes every exterior spin adjacent to the finite volume to

+ 1

. The increasing-volume limit selects the maximal infinite-volume Gibbs state.

Peierls contour

A Peierls contour is a dual-lattice boundary separating clusters of opposite Ising spins. Flipping the spins inside a contour of size

m

changes its Boltzmann weight by the factor

e^{- 2 β m}

One-dimensional Ising correlation function

For the zero-field nearest-neighbour Ising model on

Z

⟨ σ_{x} σ_{y} ⟩ = (tanh β)^{∣ x - y ∣} .

More generally, an even ordered product factors over alternating gaps.

Spontaneous magnetization ( $m (β)$ )

Spontaneous magnetization is the magnetization that remains after taking the thermodynamic limit and then sending a symmetry-breaking external field to zero.

Random-bond Ising model

A random-bond Ising model has couplings whose signs or magnitudes vary between edges according to quenched disorder. Negative bonds frustrate neighboring spins that would otherwise align.

Transverse-field Ising model

The transverse-field Ising model is a quantum spin model with competing Ising exchange and a perpendicular field, for example

H = - J \sum_{i} X_{i} X_{i + 1} - h \sum_{i} Z_{i}

. In one dimension its ordered and paramagnetic phases meet at

∣ h / J ∣ = 1

Ising-chain quantum critical point ( $g_{c} = 1$ )

For positive ferromagnetic exchange and nonnegative transverse field, the Ising-chain quasiparticle dispersion closes its gap at

g = 1

, with

ϵ_{k} \sim 2 J ∣ k ∣

. This is the quantum phase transition between the model's ordered and field-polarized regimes. The exact dispersion identifies the gapless scale; computing its order parameter additionally requires ground-state correlations.

Ising-chain Bogoliubov diagonalization ( $ϵ_{k} = 2 J 1 + g^{2} - 2 g cos k$ )

For a paired non-self-conjugate momentum, rotate the Ising-chain Nambu spinor by an angle with

cos θ = (g - cos k) / r_{k}

sin θ = - sin k / r_{k}

and

r_{k} = 1 + g^{2} - 2 g cos k

. Opposite angles at

k, - k

preserve the canonical anticommutation relations. The result is

H = \sum_{k} ϵ_{k} (γ_{k}^{†} γ_{k} - 1/2)

. Self-paired momenta have no pairing and are handled directly as number levels. Periodic spin-chain sectors additionally constrain fermion parity and allowed mode quantization.

Ising-chain ground-state energy ( $E_{0} = - \frac{1}{2} \sum_{k} ϵ_{k}$ )

The vacuum of the diagonal Bogoliubov quasiparticles has the displayed energy. In the thermodynamic bulk,

E_{0} / N = - J (2 π)^{- 1} \int_{- π}^{π} 1 + g^{2} - 2 g cos k d k

. The Hellmann–Feynman theorem then gives the field magnetization

\sum_{n} ⟨ S_{n}^{z} ⟩ = \sum_{k} J (g - cos k) / ϵ_{k}

. Sector-dependent finite-chain boundary terms must be restored when exact finite-size levels are needed.

Ising-chain quasiparticle dispersion ( $ϵ_{k} = 2 J 1 + g^{2} - 2 g cos k$ )

For nonnegative field

g

, the bulk quasiparticle gap is

2 J ∣ g - 1∣

. At

g = 0

the dispersion is flat,

2 J

; at large

g

it is

2 J g - 2 J cos k + O (J / g)

. At

g = 1

it becomes

4 J ∣ sin (k /2) ∣

, linear near the gapless point. The finite periodic spin chain has additional global excitation constraints; a single isolated domain wall is not an allowed closed-chain spin configuration.

Clock model

The clock model places a planar unit vector at every lattice site but restricts its angle to finitely many equally spaced values. Nearest-neighbour spins usually interact through the cosine of their angular difference.

Four-state clock model

The four-state clock model permits the four angles

0, π /2, π, 3 π /2

. Its finite discrete symmetry allows finite-temperature long-range order above one spatial dimension.

Classical O(n) lattice model ( $O (n)$ )

The classical O(n) lattice model assigns a unit vector

σ_{x} \in S^{n - 1}

to each lattice site and gives ferromagnetic nearest-neighbour configurations weight proportional to

exp (β \sum_{x y} σ_{x} \cdot σ_{y} + h \sum_{x} σ_{x} \cdot e)

XY model

The XY model is the infinite-state limit of the clock model: each spin is a unit vector with a continuous angle and the global symmetry is

O (2)

Ginibre inequality

For the ferromagnetic O(2) model, cosine observables with nonnegative integer coefficients are positively correlated. One form is

⟨ cos (a \cdot θ) cos (b \cdot θ)⟩ \geq ⟨ cos (a \cdot θ)⟩ ⟨ cos (b \cdot θ)⟩

for

a, b \in Z_{\geq 0}^{Λ}

, with analogous product forms.

Reflection positivity

A measure is reflection positive for a reflection

ϑ

when

E [F ϑF] \geq 0

for every observable

F

supported in one reflected half. Product measures and suitable ferromagnetic interactions satisfy this property.

Gaussian domination

Gaussian domination bounds the partition function of a reflection-positive spin model under a mean-zero external source by the partition function times a Gaussian exponential involving the inverse lattice Laplacian.

Infrared bound

The infrared bound controls the nonzero Fourier modes of a spin correlation by the inverse lattice dispersion. Its momentum integral is finite in dimension at least three and can force spontaneous magnetization at low temperature.

Phi-four lattice model ( $ϕ^{4}$ )

The phi-four lattice model is a real scalar lattice field with nearest-neighbour gradient energy and an on-site potential

g ϕ^{4} /4 + ν ϕ^{2} /2

Random-walk representation of a lattice-field covariance

The Helffer-Sjöstrand random-walk representation expresses a lattice-field covariance as the Green function of a walk in an environment determined by the Hessian of the field potential. A uniformly positive Hessian supplies killing and exponential covariance decay.

Boltzmann equation

The Boltzmann equation evolves a phase-space distribution by balancing free streaming, external forces, and collisions.

Renormalized Boltzmann solution

A weak solution framework in which a suitable nonlinear transform

β (f)

satisfies a transport equation with transformed collision source

β^{'} (f) Q (f, f)

, interpreted with the required integrability and possible framework-dependent defect terms. Renormalization addresses distributions whose raw collision term is not integrable enough to support the ordinary weak formulation. This is different from a classical smooth solution and does not by itself imply smooth global well-posedness.

Hydrodynamic limit of the Boltzmann equation

In a scaling

\partial_{t} f + v \cdot \nabla_{x} f = ε^{- 1} Q (f, f)

with small Knudsen number, the leading collision balance formally forces a local Maxwellian. The conserved moments then give the Euler equations with a monatomic ideal-gas closure. The next nonequilibrium correction produces viscous and heat-conduction terms. These substitutions are formal: rigorous limits need uniform estimates, initial and boundary conditions, and sufficient convergence.

Spatially homogeneous Boltzmann equation ( $\partial_{t} f = Q (f, f)$ )

When the velocity distribution is independent of position and there are no forces, the free transport equation contributes zero and only collisions evolve

f

. For a bounded cutoff kernel,

∥ Q (f, g) ∥_{1} \leq C ∥ f ∥_{1} ∥ g ∥_{1}

gives local existence and uniqueness by the Banach fixed-point theorem. Positivity and mass conservation bound the norm and permit global continuation in this particular model. Stronger kernels require moment or weighted estimates.

Nonlinear Boltzmann collision operator ( $Q (f, f) (v) = \int B (g, n) (f^{'} f_{}^{'} - f f_{}) d v_{*} d n$ )

For equal-mass elastic collisions, one reflection parametrization is

v^{'} = v - ((v - v_{*}) \cdot n) n

and

v_{*}^{'} = v_{*} + ((v - v_{*}) \cdot n) n

. The fixed-

n

transformation preserves pairwise momentum, kinetic energy and velocity-pair measure. A nonnegative kernel invariant under particle interchange and collision reversal yields a gain-loss operator. Its weak Boltzmann collision identity proves conservation and its entropy identity proves the Boltzmann H theorem. Unlike the Maxwell molecule collision operator, this general kernel may depend on relative speed.

Dissipation form of the linearized Boltzmann operator ( $- ⟨ h, L h ⟩_{M} = \frac{1}{4} \int BM M_{} (Δ h)^{2} d v d v_{} d n$ )

Write

f = M (1 + h)

around a fixed Maxwellian and expand the nonlinear Boltzmann collision operator to first order. The weak Boltzmann collision identity gives the nonnegative dissipation form in

L^{2} (M d v)

, and polarization proves symmetry on an appropriate common domain. Its zero modes are the collision invariants when the scattering kernel is nondegenerate. A spectral gap, and self-adjointness of an unbounded realization, require additional domain and kernel hypotheses.

Grad angular cutoff ( $\int_{S^{2}} B (g, n) d n < \infty$ )

The angular collision rate is integrable at each nonzero relative velocity, with any needed speed-growth bounds stated separately. This makes the separate gain and loss integrals available under suitable velocity integrability. Non-cutoff kernels can have infinite angular collision rate and require cancellation rather than the same naive gain-loss split.

Weak Boltzmann collision identity ( $\int Q (f, f) ψ d v = \frac{1}{2} \int B f f_{} Δ ψ d v d v_{} d n$ )

Here

Δ ψ = ψ (v^{'}) + ψ (v_{*}^{'}) - ψ (v) - ψ (v_{*})

is the pairwise collision change, not a Laplace operator. Average over the two particle labels and use collision reversal to obtain the formula. Every collision invariant has

Δ ψ = 0

, giving mass, momentum and kinetic energy conservation. A second symmetrization replaces

f f_{*} Δ ψ /2

(f^{'} f_{*}^{'} - f f_{*}) (- Δ ψ) /4

Local Maxwellian ( $M (t, x, v) = exp (a (t, x) + b (t, x) \cdot v + c (t, x) ∣ v ∣^{2}), c < 0$ )

A Gaussian velocity profile at each position, with coefficients encoding local density, mean velocity and temperature. Its logarithm is a linear combination of collision invariants, so its collision term vanishes pointwise in position. Transport need not vanish: the coefficients must also satisfy their transport equations for a local Maxwellian to solve the full Boltzmann equation.

Expanding Gaussian solution of the Boltzmann equation ( $f (t, x, v) = A exp (- α ∣ x - t v ∣^{2} - β ∣ v ∣^{2}), A, α, β > 0$ )

This spatially nonconstant local Maxwellian is constant along free characteristic curves. Its collision term vanishes by the collision invariants, so it solves the full Boltzmann equation. It is positive, rapidly decaying and integrable in phase space. Completing its velocity square gives local mean

α t x / (β + α t^{2})

and local variance proportional to

(β + α t^{2})^{- 1}

, while its spatial profile broadens with time.

Boltzmann H functional ( $H (f) = \int f lo g f d x d v$ )

The integral uses

0 lo g 0 = 0

when it is defined. For the Boltzmann equation, the Boltzmann H theorem makes it nonincreasing under suitable regularity and integrability. This functional is distinct from the microstate-counting Boltzmann entropy; the physical kinetic entropy has the opposite sign.

Boltzmann entropy dissipation ( $D (f) = \frac{1}{4} \int B (A - B_{0}) lo g (A / B_{0}) d v d v_{*} d n$ )

Put

A = f^{'} f_{*}^{'}

and

B_{0} = f f_{*}

. The weak Boltzmann collision identity applied to

lo g f

gives

H^{'} (t) = - \int D (f) d x

when transport boundary fluxes vanish. Monotonicity of the logarithm makes every integrand nonnegative. For positive sufficiently regular distributions and nondegenerate angular scattering, zero dissipation forces

lo g f

to be a collision invariant, hence

f

to be a local Maxwellian. This equality statement alone does not establish convergence rates.

Boltzmann H theorem ( $H^{'} (t) = - \frac{1}{16 π} \int (a - b) lo g (a / b) \leq 0$ )

For the normalized isotropic Maxwell molecule collision operator, put

a = f^{'} f_{*}^{'}

and

b = f f_{*}

. The integral is over position, both velocities and collision direction. The symmetrized weak identity yields the displayed derivative of the Boltzmann H functional. It is nonpositive because

(a - b) lo g (a / b) \geq 0

for positive

a, b

. Spatial transport contributes only a boundary flux, which must vanish for the total-space inequality.

Maxwell molecule collision operator ( $Q (f, f) = \frac{1}{4 π} \int (f^{'} f_{}^{'} - f f_{}) d v_{*} d σ$ )

For the normalized isotropic cutoff model, collisions redistribute the direction of relative velocity uniformly while keeping its length. With center

C = (v + v_{*}) /2

and relative speed

r = ∣ v - v_{*} ∣

, outgoing velocities are

C \pm r σ /2

. Momentum and kinetic energy are preserved, and the collision rate has no relative-speed factor. The displayed operator uses unit total angular rate. General Maxwell molecule kernels can have nonconstant angular dependence.

Collision invariant ( $ϕ (v^{'}) + ϕ (v_{}^{'}) = ϕ (v) + ϕ (v_{})$ )

A collision invariant has an unchanged pairwise sum in every allowed elastic collision. The functions

1

, each velocity coordinate, and

∣ v ∣^{2}

encode mass, momentum and energy. Symmetrized weak collision identities then give zero moments of the collision operator against these functions. Under standard regularity assumptions, the invariants form their five-dimensional linear span in three velocity dimensions.

Bobylev identity ( $Q (ξ) = \frac{1}{4 π} \int f (ξ^{+}) f (ξ^{-}) d σ - f (ξ) f (0)$ )

For the Maxwell molecule collision operator and the Fourier transform with exponent

- i ξ \cdot v

, put

ξ^{\pm} = (ξ \pm ∣ ξ ∣ σ) /2

. The angular exchange for elastic collisions, rotation invariance of spherical surface area and factorization of the two velocity integrals give the displayed formula. This identity converts the collision integral into a spherical average of products at two related frequencies.

Fourier distance of order two ( $d_{2} (f, g) = sup_{ξ \neq = 0} ∣ f (ξ) - g (ξ) ∣/∣ ξ ∣^{2}$ )

For finite nonnegative measures with equal mass and first moment and finite second moments, the constant and linear terms of their Fourier transforms cancel. Taylor's remainder bounds their difference by a constant times

∣ ξ ∣^{2}

, making this distance finite. Fourier uniqueness gives definiteness. The origin is excluded from the supremum; the quotient need not have a direction-independent limit there.

Fourier nonexpansion for Maxwell molecules ( $d_{2} (f_{t}, g_{t}) \leq d_{2} (f_{0}, g_{0})$ )

Unit-mass solutions of the isotropic Maxwell molecule collision operator with matching first moments obey this comparison. The Bobylev identity, the Fourier bound by one, and

∣ ξ^{+} ∣^{2} + ∣ ξ^{-} ∣^{2} = ∣ ξ ∣^{2}

give a scalar damped differential inequality. Its integral form and the Gronwall inequality yield nonexpansion in the Fourier distance of order two. This does not alone establish strict contraction or equilibrium convergence.

Angular exchange for elastic collisions ( $(C, r, ω, σ) \leftrightarrow (C, r, σ, ω)$ )

Write incoming relative velocity as

r ω

and outgoing relative velocity as

r σ

. For isotropic scattering, the product measure

d C r^{2} d r d ω d σ

is invariant under exchanging the two angular variables. This gives the pre/post-collision weak identities. Keeping

σ

fixed and sending only the velocities to the outgoing pair is not invertible: that map discards the incoming angular direction.

Linear Boltzmann equation ( $\partial_{t} f + v \cdot \nabla_{x} f = K_{t} f - a f$ )

A linear kinetic transport equation with velocity-integral gain

K_{t}

and nonnegative loss coefficient

a

. Its characteristic integral formulation combines the damped free-transport evolution with the Boltzmann Volterra operator. A uniform square-integrable velocity kernel yields bounded gain on

L_{x, v}^{2}

and a globally convergent Volterra series for the linear Boltzmann equation on every finite time interval.

Order preservation for the linear Boltzmann equation ( $f_{1, 0} \geq f_{2, 0} ⟹ f_{1, t} \geq f_{2, t}$ )