Critical dynamics

Critical dynamics is a section of the theory of critical behavior and statistical physics that describes the dynamic properties of a physical system at or near a critical point . It is a continuation and generalization of critical statics, allowing us to describe the values and characteristics of the system, which cannot be expressed only through the simultaneous equilibrium distribution functions . Such quantities are, for example, transport coefficients, relaxation rates, simultaneous correlation functions, response functions to time -dependent disturbances.

Like all statistical physics , critical dynamics deals with a huge or even infinite number of degrees of freedom . Various stochastic (random) processes are inherent in the development of such systems over time: thermal motion and collision of molecules in a gas system, reorientation of lattice spins in a solid, and the appearance and interaction of turbulent vortices in a fluid flow. The formulation and solution of such problems is carried out using the formalism of quantum field theory , which was originally created for the needs of high-energy physics and elementary particles. The stochasticity of processes is modeled by introducing into the dynamic equations an additional random term - “noise” with a known (usually Gaussian ) distribution.

Short description of the system

Stochastic Dynamics Problem Setting

Marking for $x$ ${\ displaystyle x}$ $x$ set of spatial coordinates and system indices, for $\phi (x)$ ${\ displaystyle \ phi (x)}$ $\phi(x)$ the entire set of fields in the system, we can write the standard formulation of the problem of stochastic dynamics.

 $\partial _{t}\phi (x)=U(x;\phi )+\eta (x),\ \ \ <{\hat {\eta }}(x){\hat {\eta }}(x')>=D(x,x')$             $\partial$         $t$           $ϕ$       $($       $x$       $)$       $=$       $U$       $($       $x$       $;$       $ϕ$       $)$       $+$       $η$       $($       $x$       $)$       $,$             $<$             $η$       $^$             $($       $x$       $)$             $η$       $^$             $($         $x$       $'$         $)$       $> =$       $D$       $($       $x$       $,$         $x$       $'$         $)$           ${\ displaystyle \ partial _ {t} \ phi (x) = U (x; \ phi) + \ eta (x), \ \ \ <{\ hat {\ eta}} (x) {\ hat {\ eta }} (x ')> = D (x, x')}$     $\partial _{t}\phi (x)=U(x;\phi )+\eta (x),\ \ \ <{\hat {\eta }}(x){\hat {\eta }}(x')>=D(x,x')$

U here is the given t-local functional,

\eta

{\ displaystyle \ eta}

\eta

- a random external force that simulates all rapidly changing processes in the system. For it, a Gaussian distribution with a zero mean and a given correlator D is assumed. The delay condition and some boundary conditions are also satisfied, which are usually taken to be zero at times

t->-\infty

{\ displaystyle t -> - \ infty}

t->-\infty

This is the most general form of the equation of evolution in problems of stochastic dynamics. Of course, not with any choice of the functional U and the correlator D it will have a simple solution.

Below are a few examples of stochastic dynamics problems.

Brownian motion

We write the equations for Brownian motion in the language of stochastic dynamics:

$\partial _{t}r_{i}(t)=\eta _{i}(t),\ \ \ <{\hat {\eta }}_{i}(t){\hat {\eta }}_{k}(t')>=2\alpha \delta _{ik}\delta (t-t')$ ${\ displaystyle \ partial _ {t} r_ {i} (t) = \ eta _ {i} (t), \ \ \ <{\ hat {\ eta}} _ {i} (t) {\ hat { \ eta}} _ {k} (t ')> = 2 \ alpha \ delta _ {ik} \ delta (t-t')}$

Here $\phi (x)\equiv r_{i}(t)$ ${\ displaystyle \ phi (x) \ equiv r_ {i} (t)}$ , U = 0, constant $\alpha$ ${\ displaystyle \ alpha}$ carries the meaning of diffusion coefficient.

Navier-Stokes equation

The dynamic Navier-Stokes equation can also be formulated in this language. The critical tasks for the equation will be the description of turbulence , including developed turbulence (for systems with large Reynolds numbers), the construction of the distribution function of the vortices along the wave vector (in the Fourier representation of the velocity field), and the verification of Kolmogorov's phenomenological theory.

 $\partial _{t}\phi _{i}(x)=\nu \Delta \phi _{i}(x)-\partial _{i}P-\phi _{j}\partial _{j}\phi _{i}$             $\partial$         $t$             $ϕ$         $i$           $($       $x$       $)$       $=$       $ν$       $Δ$         $ϕ$         $i$           $($       $x$       $)$       $-$         $\partial$         $i$           $P$       $-$         $ϕ$         $j$             $\partial$         $j$             $ϕ$         $i$               ${\ displaystyle \ partial _ {t} \ phi _ {i} (x) = \ nu \ Delta \ phi _ {i} (x) - \ partial _ {i} P- \ phi _ {j} \ partial _ {j} \ phi _ {i}}$

 $\partial _{i}\phi _{i}=0\ \$             $\partial$         $i$             $ϕ$         $i$           $=$       $0$               ${\ displaystyle \ partial _ {i} \ phi _ {i} = 0 \ \}$       (transverse condition)

Here

\phi

{\ displaystyle \ phi}

- incompressible vector velocity field,

\nu

{\ displaystyle \ nu}

- kinematic viscosity, p - pressure.

Langevin type problems

In the class of stochastic dynamics problems, a narrower class of critical dynamics problems is traditionally distinguished, in which additional conditions are imposed on the fields under consideration and the form of the functional U (t-local functional standing on the right side of the dynamic equation for fields). Firstly, as a set of system fields, a set of fields corresponding to the so-called soft mods. A soft mode is any quantity whose large-scale fluctuations slowly relax, i.e., in the momentum representation, the relaxation rate of fluctuations with a given wave vector k tends to zero at $k->0$ ${\ displaystyle k-> 0}$ . For example, the field of the order parameter near the critical point is itself always a soft mode. Secondly, the functional U will be the variational derivative of the static action. We write the corresponding statement of the problem:

$\partial _{t}\phi _{a}=(\alpha _{ab}+\beta _{ab}){\frac {\delta S^{st}(\phi )}{\delta \phi _{b}}}+\eta _{a},\ \ \ <{\hat {\eta }}_{a}(x){\hat {\eta }}_{b}(x')>=2\alpha _{ab}\delta (x-x')$ ${\ displaystyle \ partial _ {t} \ phi _ {a} = (\ alpha _ {ab} + \ beta _ {ab}) {\ frac {\ delta S ^ {st} (\ phi)} {\ delta \ phi _ {b}}} + \ eta _ {a}, \ \ \ <{\ hat {\ eta}} _ {a} (x) {\ hat {\ eta}} _ {b} (x ' )> = 2 \ alpha _ {ab} \ delta (x-x ')}$

$\alpha _{ab}$ ${\ displaystyle \ alpha _ {ab}}$ here called the Onsager coefficient, $\beta {ab}$ ${\ displaystyle \ beta {ab}}$ - intermode communication.

The following conditions are fulfilled for them:

$\alpha _{ab}=\alpha _{ba}$ ${\ displaystyle \ alpha _ {ab} = \ alpha _ {ba}}$ , i.e., the Onsager coefficient is symmetric (this can easily be understood from the fact that the correlator of perturbations of random forces is symmetric by definition)

$\beta _{ab}=-\beta _{ba}$ ${\ displaystyle \ beta _ {ab} = - \ beta _ {ba}}$

The justification of the intermode coupling properties is carried out using the Fokker-Planck equation .

Thus, the formulation of one or another critical dynamics problem corresponds to the assignment of a set of fields describing the system, the Onsager coefficient, and intermode coupling. The following is a list of the most common and studied models.

Critical Dynamics Models

Following the classic article [Hohenberg, Halperin], we give a standard list of models of critical dynamics. They all correspond to static $O_{n}-\phi ^{4}$ ${\ displaystyle O_ {n} - \ phi ^ {4}}$ -models for the order parameter field, the action in these models will be given explicitly.

Static action $O_{n}-\phi ^{4}$ ${\ displaystyle O_ {n} - \ phi ^ {4}}$ models for the n-component field $\psi$ ${\ displaystyle \ psi}$ is an

 $S^{st}(\psi )=-(\partial \psi )^{2}/2-\tau \psi ^{2}/2-v_{1}(\psi ^{2})^{2}/24$             $S$         $s$       $t$           $($       $ψ$       $)$       $=$       $-$       $($       $\partial$       $ψ$         $)$         $2$             $/$         $2$       $-$       $τ$         $ψ$         $2$             $/$         $2$       $-$         $v$         $one$           $($         $ψ$         $2$             $)$         $2$             $/$         $24$           ${\ displaystyle S ^ {st} (\ psi) = - (\ partial \ psi) ^ {2} / 2- \ tau \ psi ^ {2} / 2-v_ {1} (\ psi ^ {2}) ^ {2} / 24}$

Models A and B

A and B are relaxation models, i.e., the intermode coupling (the antisymmetric part of the corresponding matrix) is equal to zero.Model A describes an anisotropic ferromagnet with a one-component non-conserved field of the order parameter, for which the projection of magnetization onto one of the coordinate axes is considered in the physical system;Model B describes a uniaxial ferromagnet with a single-component conserved field of the order parameter, for which the projection of magnetization onto one of the coordinate axes stands in the physical system.Model A:

 $\phi =\{\psi \},S^{st}(\psi ),\alpha _{\psi }=\lambda _{\psi }$           $ϕ$       $=$       ${$       $ψ$       $}$       $,$         $S$         $s$       $t$           $($       $ψ$       $)$       $,$         $α$         $ψ$           $=$         $λ$         $ψ$               ${\ displaystyle \ phi = \ {\ psi \}, S ^ {st} (\ psi), \ alpha _ {\ psi} = \ lambda _ {\ psi}}$       ,

Where

\lambda _{a}\equiv const>0\ \ \forall a

{\ displaystyle \ lambda _ {a} \ equiv const> 0 \ \ \ forall a}

Model B:

 $\phi =\{\psi \},S^{st}(\psi ),\alpha _{\psi }=-\lambda _{\psi }\cdot \partial ^{2}$           $ϕ$       $=$       ${$       $ψ$       $}$       $,$         $S$         $s$       $t$           $($       $ψ$       $)$       $,$         $α$         $ψ$           $=$       $-$         $λ$         $ψ$           $\cdot$         $\partial$         $2$               ${\ displaystyle \ phi = \ {\ psi \}, S ^ {st} (\ psi), \ alpha _ {\ psi} = - \ lambda _ {\ psi} \ cdot \ partial ^ {2}}$

From the point of view of the formal statement of the model, A and B differ, therefore, only by maintaining the field of the order parameter.

Models C and D

Models C and D are also purely relaxation. They are generalizations of models A and B to the case of energy conservation; they introduce an additional persistent scalar field

m(x)=<{\hat {E}}(x)>

{\ displaystyle m (x) = <{\ hat {E}} (x)>}

describing temperature fluctuations.

Model C:

 $\phi =\{\psi ,m\}$           $ϕ$       $=$       ${$       $ψ$       $,$       $m$       $}$           ${\ displaystyle \ phi = \ {\ psi, m \}}$       , where m is the additional conserved one-component field

 $S^{st}(\phi )=S^{st}(\psi )-m^{2}/2-v_{2}m\psi ^{2}/2$             $S$         $s$       $t$           $($       $ϕ$       $)$       $=$         $S$         $s$       $t$           $($       $ψ$       $)$       $-$         $m$         $2$             $/$         $2$       $-$         $v$         $2$           $m$         $ψ$         $2$             $/$         $2$           ${\ displaystyle S ^ {st} (\ phi) = S ^ {st} (\ psi) -m ^ {2} / 2-v_ {2} m \ psi ^ {2} / 2}$

 $\alpha _{\psi }=\lambda _{\psi };\ \ \ \alpha _{m}=-\lambda _{m}\cdot \partial ^{2}$             $α$         $ψ$           $=$         $λ$         $ψ$           $;$               $α$         $m$           $=$       $-$         $λ$         $m$           $\cdot$         $\partial$         $2$               ${\ displaystyle \ alpha _ {\ psi} = \ lambda _ {\ psi}; \ \ \ \ alpha _ {m} = - \ lambda _ {m} \ cdot \ partial ^ {2}}$

Model D:

 $\phi =\{\psi ,m\}$           $ϕ$       $=$       ${$       $ψ$       $,$       $m$       $}$           ${\ displaystyle \ phi = \ {\ psi, m \}}$       , where m is the additional conserved one-component field

 $S^{st}(\phi )=S^{st}(\psi )-m^{2}/2-v_{2}m\psi ^{2}/2$             $S$         $s$       $t$           $($       $ϕ$       $)$       $=$         $S$         $s$       $t$           $($       $ψ$       $)$       $-$         $m$         $2$             $/$         $2$       $-$         $v$         $2$           $m$         $ψ$         $2$             $/$         $2$           ${\ displaystyle S ^ {st} (\ phi) = S ^ {st} (\ psi) -m ^ {2} / 2-v_ {2} m \ psi ^ {2} / 2}$

 $\alpha _{\psi }=-\lambda _{\psi }\cdot \partial ^{2};\ \ \ \alpha _{m}=-\lambda _{m}\cdot \partial ^{2}$             $α$         $ψ$           $=$       $-$         $λ$         $ψ$           $\cdot$         $\partial$         $2$           $;$               $α$         $m$           $=$       $-$         $λ$         $m$           $\cdot$         $\partial$         $2$               ${\ displaystyle \ alpha _ {\ psi} = - \ lambda _ {\ psi} \ cdot \ partial ^ {2}; \ \ \ \ alpha _ {m} = - \ lambda _ {m} \ cdot \ partial ^ {2}}$

Again, from the point of view of the formal formulation of the model, C and D differ only in the conservation of the order parameter field.

Literature

Vasiliev A.N. Quantum-field renormalization group in the theory of critical behavior and stochastic dynamics. - SPb. : publishing house of PNPI, 1998.
Hohenberg PC Halperin BI Theory of dynamic critical phenomena, Rev. Mod. Phys, Vol. 49, No. 3, July 1977, pp 435-475

Notes

Links

MSR formalism