# Symmetry Breaking in Stochastic Dynamics and Turbulence

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## Abstract

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## 1. Introduction

- Systems with a Hermitian Hamiltonian, whose stationary states are described by Gibbs–Boltzmann distribution. Note that at the beginning they happen to be in a state far from the stationary (equilibrium) state. The dynamic description of such systems is obtained directly from static formulations. Examples include the Landau–Ginzburg equation for time evolution of local magnetization, kinetic Ising model, and models A-H for various models of critical dynamics [16]. All these equations are specific realizations of a rather general Langevin equation [5].
- Systems without Hermitian Hamiltonian or without Hamiltonian description at all, which in general do not need to have a stationary state. The detailed balance condition is not satisfied for them, which implies that Einstein relation between thermal fluctuations and friction forces cannot be stated. Typical examples of such systems cover: fluid in turbulent state, irreversible chemical processes, surface growing models, etc. Other approaches to such systems have to be used via quite general stochastic differential equation, which can be considered as an extension of a Langevin equation or using a master equation [17]. The former equation is suggested for some macroscopic quantity. Neglect of microscopic degrees of freedom is replaced by an introduction of random force. Then, according to underlying physical observations, properties of random force have to be specified. The latter approach is probably more fundamental, but also more difficult to handle.

## 2. Field-Theoretic Formulation

## 3. Stochastic Approach to Turbulence

- The rapid-change model corresponding to the limit ${u}_{10}\to \infty ,{g}_{10}^{\prime}\equiv {g}_{10}/{u}_{10}=const$. Then, the kernel function becomes$${D}_{v}({\omega}_{k},\mathit{k})\propto {g}_{10}^{\prime}{D}_{0}{k}^{-d-\epsilon +\eta}.$$The velocity correlator is obviously $\delta $—correlated in the time variable.
- The frozen velocity field arising in the limit ${u}_{10}\to 0$, in which the kernel function corresponds to$${D}_{v}({\omega}_{k},\mathit{k})\propto {g}_{0}{D}_{0}^{2}\pi \delta \left({\omega}_{k}\right){k}^{2-d-\epsilon}.$$
- The turbulent advection, for which $\epsilon =2\eta =8/3$. This choice mimics properties of the fully developed turbulence and yields well-known Kolmogorov scaling [23].

## 4. Renormalization Group Analysis

- For any dynamic model in Equation (1), all 1PI Green functions containing only the original fields $\varphi $ are proportional to the closed loops of step functions, hence they vanish, and thus do not generate counterterms.
- If for some reason several external momenta or frequencies occur as an overall factor in all the Feynman diagrams of a particular Green function, the real degree of divergence ${\delta}^{\prime}$ is less than $\delta \equiv {d}_{\mathsf{\Gamma}}(\epsilon =0)$ by the corresponding number of units.
- Sometimes the divergences formally allowed by dimensionality are absent due to symmetry restrictions, for instance, the Galilean invariance of the fully developed turbulence [31] restricts the form of possible counterterms.
- Nonlocal terms of the model are not renormalized.

## 5. Composite Operators and Operator Product Expansion

## 6. Schwinger Equations and Conservation Laws

## 7. Ward Identities and Galilean Invariance

## 8. Symmetry Restoration

## 9. Parity Breaking in Magnetohydrodynamic Turbulence

- ${\mathcal{S}}_{{b}^{\prime}bv}$: ${b}_{i}^{\prime}(-{v}_{j}{\partial}_{j}{b}_{i}+A{b}_{j}{\partial}_{j}{v}_{i})={b}_{i}^{\prime}{v}_{j}{V}_{ijl}{b}_{l}$,
- ${\mathcal{S}}_{{v}^{\prime}vv}$:$-{v}_{i}^{\prime}{v}_{j}{\partial}_{j}{v}_{i}={v}_{i}^{\prime}{v}_{j}{W}_{ijl}{v}_{l}/2,$
- ${\mathcal{S}}_{{v}^{\prime}bb}$:${v}_{i}^{\prime}{b}_{j}{\partial}_{j}{b}_{i}={v}_{i}^{\prime}{v}_{j}{U}_{ijl}{v}_{l}/2,$

## 10. Effect of Strong Anisotropy

- The most relevant anisotropy correction is of order $\left({\alpha}_{1,2}\right)$ for $p\ne 0$ and $\left({\alpha}_{1,2}^{2}\right)$ for $p=0$. This means that ${\gamma}^{*}[N,0]$ are anisotropy independent in the linear approximation.
- This leading contribution depends on ${\alpha}_{1,2}$ only through the combination ${\alpha}_{3}\equiv 2{\alpha}_{1}+d{\alpha}_{2}$.

- If the mixed correlator $\langle vf\rangle $ is missing, the odd structure functions vanish. At the same time, the contributions to even functions are given only by the operators with even values of ${N}^{\prime}$. Only contributions with $p=0$ remain in the isotropic case (${\alpha}_{1,2}=0$) [104,108]. When anisotropy is present, ${\alpha}_{1,2}\ne 0$, the operators with $p\ne 0$ assume nonvanishing means, and their dimensions $\Delta [{N}^{\prime},p]$ enter the right hand side of Equation (237).
- The most relevant term of the small $r/L$ behavior is obviously related to the contribution with the smallest value of the exponent $\Delta [{N}^{\prime},p]$. Now, we employ the hierarchy relations in Equations (229) and (230), which hold for ${\alpha}_{1,2}=0$ and consequently remain valid at least for ${\alpha}_{1,2}\ll 1$. We conclude that, for sufficiently weak anisotropy, the leading term in (237) is given by the dimension $\Delta [N,0]$ for any ${S}_{N}$. In all particular cases considered, this hierarchy remains for finite values of the anisotropy parameters as well. The contribution with $\Delta [N,0]$ stays leading for such N and ${\alpha}_{1,2}$.
- The introduction of the mixed correlator $\langle vf\rangle \propto n\delta (t-{t}^{\prime})\phantom{\rule{0.166667em}{0ex}}{C}^{\prime}(r/L)$ explicitly violates the evenness in $n$ and generates non-vanishing odd functions ${S}_{2n+1}$ and leads to to the contributions with odd ${N}^{\prime}$ to the expansion in Equation (237) for even functions. If the relations in Equations (229) and (230) are satisfied, in even functions, the leading term is still given by the contribution with $\Delta [N,0]$. If the relations in Equation (231) are fulfilled, in the odd function ${S}_{2n+1}$, the leading term is given by the dimension $\Delta [2n,0]$ for $n<(d+2)/4$ and by $\Delta [2n+1,1]$ for $n>(d+2)/4$. Let us note that, for the model with an imposed gradient, for all n, the leading terms of ${S}_{2n+1}$ are given by the dimensions $\Delta [2n+1,1]$ [34,36].

## 11. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NS | Navier–Stokes |

RG | Renormalization Group |

SDE | Stochastic Differential Equation |

UV | Ultraviolet |

IR | Infrared |

OPE | Operator Product Expansion |

1PI | one-particle irreducible |

MHD | Magnetohydrodynamics |

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**Figure 1.**Nontrivial propagators for the model in Equation (24).

**Figure 2.**Interaction vertex responsible for the nonlinear interactions between velocity fluctuations in the model in Equation (24). Momentum k on the right hand side corresponds to the inflowing momentum of the auxiliary field ${v}^{\prime}$.

**Figure 3.**Feynman rules for the model in Equation (48).

**Figure 4.**Graphical representation of all interaction vertices of the model related velocity non-linearities of the action (166).

**Figure 5.**Graphical representation of all propagators of the model given by the quadratic part of the action (166).

**Figure 6.**Graphical representation of all Feynman diagrams for two-point one-irreducible Green functions of the action (176). Graphs ${\mathsf{\Gamma}}_{1},\dots ,{\mathsf{\Gamma}}_{4}$ represent perturbation expansion for ${\mathsf{\Gamma}}_{{v}^{\prime}v}$ function, and ${\mathsf{\Gamma}}_{5},\dots ,{\mathsf{\Gamma}}_{8}$ for ${\mathsf{\Gamma}}_{{b}^{\prime}b}$ function.

**Figure 7.**Graphical representation of all one-loop Feynman diagrams forces for one-irreducible Green function ${\mathsf{\Gamma}}_{{v}^{\prime}bb}$.

**Figure 8.**Construction of the proper orthonormal basis for a sought solution of linearized MHD equations.

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Hnatič, M.; Honkonen, J.; Lučivjanský, T. Symmetry Breaking in Stochastic Dynamics and Turbulence. *Symmetry* **2019**, *11*, 1193.
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Hnatič M, Honkonen J, Lučivjanský T. Symmetry Breaking in Stochastic Dynamics and Turbulence. *Symmetry*. 2019; 11(10):1193.
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**Chicago/Turabian Style**

Hnatič, Michal, Juha Honkonen, and Tomáš Lučivjanský. 2019. "Symmetry Breaking in Stochastic Dynamics and Turbulence" *Symmetry* 11, no. 10: 1193.
https://doi.org/10.3390/sym11101193