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For additive Gaussian white noise and a time-even order parameter, a midpoint discretization of the Onsager–Machlup functional evaluates drift derivatives at the midpoint configurations. Reversing a trajectory visits the same midpoints in reverse order, so the noise-to-path Jacobian determinant is invariant under reversal. It may depend on the trajectory, and must be distinguished from the constant normalization of the noise measure.

For nonconserved order-parameter dynamics with $\dot{p}=-\Gamma \mu [p]+f$, the factor at one time step, apart from a path-independent power of the step size, isHere the fields have first been restricted to a finite spatial grid, and ${\mathcal{H}}_n$ is the Hessian matrix of the free energy.

For mixed conserved and nonconserved order-parameter dynamics, the required noises for a joint configuration and flux trajectory arewhere $\mathsf{D},\mathsf{G}$ represent the divergence and gradient, and $\Delta =\mathsf{DG}$ represents the Laplacian. The Jacobian matrix isTaking its Schur complement gives the joint factorFor periodic boundary conditions, discretizing the two spatial operators compatibly gives $\mathsf{D}=-{\mathsf{G}}^{\mathsf{T}}$, so $\Gamma I-M\Delta$ is the positive relaxation operator when $\Gamma ,M>0$. Reversal leaves ${\mathcal{H}}_n$ unchanged and reverses the sign of the flux. The product of the joint factors is consequently the same for both histories, justifying its cancellation from the joint path probability of an order parameter and its flux. The functional chain rule used in the action ratio holds in the continuum limit of this midpoint convention.

A vector order parameter has several components transforming together as a vector under spatial rotations. A polar order parameter is an example. Its functional derivative $\delta F/\delta \mathbf{p}$ has one component for each component of the order parameter.

Vitaly Ginzburg and Lev Landau described superconductivity through a spatially varying complex order parameter coupled to an electromagnetic vector potential. Their Ginzburg-Landau theory balances the cost of changing the order parameter with magnetic-field energy and the tendency to form a superconducting state. It predicts characteristic coherence and penetration lengths and supports analysis of surfaces, currents and nonuniform states. The theory is phenomenological: it can organize measurable behavior near a transition without first deriving the microscopic pairing interaction.^[ref]

A physicist who headed the Lebedev institute’s theoretical physics department from 1971 to 1988.^[ref]

Near a renormalization-group fixed point, expand a difference between two nearby models in eigenvectors of the stability matrix. An irrelevant operator direction has $y_i<0$, so its dimensionless coefficient shrinks under increasing length factor $b$. A relevant operator direction grows and must be tuned to study the same critical behavior. This is a mechanism for losing microscopic detail, not an assertion that every parameter disappears. Critical temperature, individual amplitudes and regular contributions to free energy can remain material dependent.

For the solvable example, write $A=ϵ/g_0-1$. Its exact solution gives $g-ϵ=-ϵA{b}^{-ϵ}+O(b^{-2ϵ})$. At the thermal stopping scale $b_{\ast }=|t_0{|}^{-1/y_t}$ this is of order $|t_0{|}^{ϵ/y_t}$. If the stopped correlation length is smooth in $g$, the leading correction to its power law has that order. Thus two microscopic couplings can yield the same limiting critical exponent, yet approach it at different rates.

This mechanism makes universality useful: one tractable representative can predict critical exponents and normalized scaling forms for many systems in its universality class. A liquid-gas critical point and a scalar magnetic transition can share a long-distance description even though their microscopic constituents differ. The scope must be checked: spatial dimension, order parameter symmetry, interaction range, and extra soft variables can change the renormalization-group fixed point.

In his lecture, Kenneth Geddes Wilson connects limiting renormalization-group fixed points with common critical behavior and states the limits of his elementary truncation.^[ref]

The XY model assigns a planar unit spin ${\mathbf{s}}_i=(\cos {\theta }_i,\sin {\theta }_i)$ to each lattice site, with a continuous angle ${\theta }_i$. Its usual Hamiltonian is $H=-J{\sum }_{⟨ij⟩}\cos ({\theta }_i-{\theta }_j)$. For $J>0$, alignment lowers energy, but a common rotation of all spins leaves every angle difference and hence the energy unchanged.

This continuous orthogonal group $O(2)$ symmetry distinguishes it from a finite-state clock model. Small angular variations produce a spin wave description, while large windings can introduce vortices. The XY model is useful for studying collective phase coherence as well as planar magnetism because the angle can represent an order parameter's phase rather than a literal magnetic arrow. Dimension, interaction range and temperature remain essential when deciding which kind of order the model can sustain.