**3. Sub-fluid physics as noise: A stochastic field theory for the MHD**

The metriplectic theory of the MHD discussed in § 2 clarifies how the dissipative part of the dynamics must be attributed to the presence of *statistically treated* degrees of freedom, through their entropy. On the one hand, the metriplectic MHD gives a role to the statistics of the medium properties; on the other hand, local equilibrium and space-timesmoothness of field variables are still assumed. In the sub-fluid model presented in this paragraph, the statistical nature of the microscopic degrees of freedom is cast into a form going beyond the local equilibrium condition. In particular, strong reference to plasma ISCs is made.

Plasma ISC dynamics resembles more closely a quantum transition than a classical evolution: the idea presented here is that localized occurrence of big fluctuations in the medium probably initiate and determine those quantum-like transitions of the variables **B** and **V**. If the fluctuations of the medium are treated as *probabilistic stirring forces*, or *noises*, a totally new scenario appears.

The formalism turning those considerations into a mathematical theory was introduced in Materassi & Consolini (2008); then, an application of it to the visco-resistive reduced MHD in 2 dimensions was obtained in Materassi (2009).

Let's consider the resistive incompressible MHD equations:

$$\begin{cases} \begin{aligned} \label{10} \mathcal{C}\_{t} \mathcal{B}^{i} &= \mathcal{B}^{j} \mathcal{O}\_{j} \mathcal{V}^{i} - \mathcal{V}^{j} \mathcal{O}\_{j} \mathcal{B}^{i} - \mathcal{E}^{j \text{jk}} \mathcal{O}\_{j} \left( \mathcal{L}\_{k \text{h}} \mathcal{J}^{\text{h}} \right) \\\\ \mathcal{O}\_{t} \mathcal{V}^{i} &= -\mathcal{V}^{j} \mathcal{O}\_{j} \mathcal{V}^{i} + \frac{I\_{j}}{\rho} \mathcal{B}\_{k} \varepsilon^{j \text{k}i} - \frac{\mathcal{E}^{i} p}{\rho} \end{aligned} \tag{14}$$

44 Topics in Magnetohydrodynamics

opposite of the homogeneous temperature the plasma reaches at the equilibrium. Other equilibria with non trivial magnetic or velocity fields can in principle be obtained by considering Casimir constants other than the entropy, and a different metric bracket, or

suitable physical quantities. Moreover, the boundary conditions for the system to work in this way must be such that all the fields behave "suitably" at the space infinity. All the results are obtained for a *visco-resistive isolated plasma*: indeed, all the algebraic relationships invoked hold if **V**, **B**, *ρ* and *s* show suitable boundary conditions, rendering visco-resistive

Such metriplectic formulation conserves, in addition to the energy *H*, also the total linear momentum **P**, the total angular momentum **L** and the generator of Galileo's boosts **G**, which

33 3

*d x*,, . *d x tdx* **P V L xV G xV**

About these quantities **P**, **L** and **G**, it should be stressed that, besides modifying the scheme with other quantities conserved in the ideal limit, more interesting equilibria than (13) may be identified by conditioning the extremization of *F* to the initial finite values of the Galilean

The metriplectic theory of the MHD discussed in § 2 clarifies how the dissipative part of the dynamics must be attributed to the presence of *statistically treated* degrees of freedom, through their entropy. On the one hand, the metriplectic MHD gives a role to the statistics of the medium properties; on the other hand, local equilibrium and space-timesmoothness of field variables are still assumed. In the sub-fluid model presented in this paragraph, the statistical nature of the microscopic degrees of freedom is cast into a form going beyond the local equilibrium condition. In particular, strong reference to plasma

Plasma ISC dynamics resembles more closely a quantum transition than a classical evolution: the idea presented here is that localized occurrence of big fluctuations in the medium probably initiate and determine those quantum-like transitions of the variables **B** and **V**. If the fluctuations of the medium are treated as *probabilistic stirring forces*, or *noises*, a

The formalism turning those considerations into a mathematical theory was introduced in Materassi & Consolini (2008); then, an application of it to the visco-resistive reduced MHD

> , *i ii h <sup>j</sup> j ijk t j j j kh*

*B BV V B J*

*j i i j jki t jk*

*<sup>J</sup> <sup>p</sup> V VV B*

*i*

 

 

(14)

**3. Sub-fluid physics as noise: A stochastic field theory for the MHD** 

 

*F* = 0 onto some manifold of constant value for

simply by constraining the condition

MHD a "complete system".

transformation generators.

totally new scenario appears.

in 2 dimensions was obtained in Materassi (2009).

Let's consider the resistive incompressible MHD equations:

are defined by:

ISCs is made.

(the choice of incompressible plasma is done for reasons to be clarified later). *ζ* is the resistivity tensor and *p* is the plasma pressure. The dynamical variables are the fields **V** and **B**. The viscosity *ν* is assumed to be zero. The form of *ζ* and *p*, and of the mathematical relationships among them (necessary to close the system (14)), depend on the microdynamics of the medium. Usually, constitutive hypotheses provide the information on the microscopic nature of the medium (Kelley, 1989). When the (at least local) thermodynamic equilibrium is assumed, the constitutive hypotheses read something like:

$$
\zeta' = \zeta'(T, \dots) \quad , \quad \Phi(p, T) = 0,\tag{15}
$$

being *T* the local temperature field. Then, some heat equation is invoked for *T*, requiring other constitutive hypotheses about the specific heat of the plasma.

The aforementioned procedure will only give *ζ* and *p* regular quasi-everywhere. Instead, in the sub-fluid approach presented here, irregularities of *ζ* and *p* are explicitly considered by stating that these local quantities are *stochastic fields*, and by assigning their probability density functions (PDF). The probabilistic nature of the terms *ζ* and *p* will be naturally transferred to **B** and **V** through a suitable SFT. The following vector quantities are defined

$$\Delta^i = -\varepsilon^{ijk}\hat{\sigma}\_j\left(\mathcal{L}\_{kh}I^h\right), \quad \Delta^i = \frac{f^i}{\rho}, \quad \Theta^i = -\frac{\hat{\sigma}^i p}{\rho} \, : \tag{16}$$

these **Ξ**, **Δ**, and **Θ** are considered as *stochastic stirring forces*, and their probability density functional is assigned as some *Q*[**Ξ**,**Δ**,**Θ**]. The resistive MHD equations are then re-written as the following *Langevin field equations*:

$$\begin{cases} \begin{aligned} \left\| \mathcal{C}\_t B^i = B^j \mathcal{C}\_j V^i - V^j \mathcal{C}\_j B^i + \Xi^i \end{aligned} \right\|\_2 \\ \begin{aligned} \left\| \mathcal{C}\_t V^i = -V^j \mathcal{C}\_j V^i + \Delta\_j B\_k \varepsilon^{jki} + \Theta^i \end{aligned} \right\|\_2 \\ \begin{aligned} \left\| \left(\Xi\_\nu \Delta, \Theta\right) \approx & Q \left[ \Xi\_\nu \Delta, \Theta \right] .\end{aligned} \end{cases} \tag{17}$$

This scheme, clearly, is not self-consistent because the PDF of the noise terms must be assigned *a priori*, as the outcome of a microscopic dynamics not included in this treatment and not predictable by it. Plasma microscopic physics will enter through some PDF *P*dyn[*ζ*,*p*]: as far as *P*dyn[*ζ*,*p*] keeps trace of the plasma complex dynamics, this represents a (rather general) way to provide constitutive hypotheses. Then, the positions (16) are used to construct mathematically the passage:

$$\left(\left(\boldsymbol{\zeta},\boldsymbol{p}\right)\stackrel{\text{iid}}{\approx}P\_{\text{dyn}}\left[\boldsymbol{\zeta},\boldsymbol{p}\right]\quad\Rightarrow\quad\left(\boldsymbol{\Xi},\boldsymbol{\Delta},\boldsymbol{\Theta}\right)\stackrel{\text{iid}}{\approx}Q\left[\boldsymbol{\Xi},\boldsymbol{\Delta},\boldsymbol{\Theta}\right].$$

A closed form for *Q*[**Ξ**,**Δ**,**Θ**] should be obtained consistently with any microscopic dynamical theory of the ISC plasma, from the very traditional equilibrium statistical mechanics to the fractional kinetics reviewed in Zaslavsky (2002).

Due to the presence of the stochastic terms **Ξ**, **Δ**, and **Θ** two important things happen: first of all, from each set of initial conditions, *many possible evolutions* of **B** and **V** develop according to

Sub-Fluid Models in Dissipative Magneto-Hydrodynamics 47

The quantity *L*0(**Ω**,**Π**,**B**,**V**) is interpreted as the part of the Lagrangian of the SFT not containing noise terms. *L*0 shows only space- and time-local terms, always: as it is stressed in Chang (1999), the integration of the noise term *C*[**Ω**,**Π**,**B**,**V**;*t*0,*t*) brings terms in *L* that are non-local in space and in time, due to the self- and mutual correlations of noises. Those

**Ω Π B V**

, ,,

*idL d x*

The form of *Q*[**Ξ**,**Δ**,**Θ**], hence of *C*[**Ω**,**Π**,**B**,**V**;*t*0,*t*), may render the SFT long-range correlated and with a finite memory: these conditions of the ISC plasmas described by such a SFT is what encourages people to work through the techniques of *dynamical renormalization group* (Chang et al., 1978). Possibly, the stochastic momenta may be eliminated, so that one obtains

Once *W*[**B**,**V**;*t*0,*t*) has been obtained, the calculation of processes in which the magnetized plasma changes arbitrarily, from an initial configuration (**B**(*t*0),**V**(*t*0)) = (**B**i,**V**i) to a final one (**B**(*t*),**V**(*t*)) = (**B**f,**V**f), may be done, for any time interval (*t*0,*t*): the rate of such transitions

ii ff 0i 0i

As a further development of Materassi & Consolini (2008), a complete representation *à la Feynman* of such processes is to be derived from the SFT, with a suitable perturbative theory

In order to arrive to a closed expression for a stochastic action at least in one example case, hereafter a toy model is reported, in which **Ξ**, **Δ** and **Θ** are assumed to be *Gaussian processes without any memory, and δ-correlated in space*. This hypothesis is surely over-simplifying for a plasma in ISC, since there are experimental results stating the presence of non-Gaussian distributions (Yordanova et al., 2005), and also of memory effects (Consolini et al., 2005). Nevertheless, the Gaussian example is of some use in illustrating the SFT at hand, because a Gaussian shape for *Q*[**Ξ**,**Δ**,**Θ**] allows for the full integration of *C*[**Ω**,**Π**,**B**,**V**;*t*0,*t*), and the explicit calculation of *W*[**B**,**V**;*t*0,*t*) from *A*[**Ω**,**Π**,**B**,**V**;*t*0,*t*) in (21). The probability density functional *Q*[**Ξ**,**Δ**,**Θ**] is obtained via a *continuous product* out of distributions of the local values of the fields **Ξ**, **Δ** and **Θ** of Gaussian nature; for instance, the PDF of the local variable **Ξ**(**x**,*t*) reads:

<sup>2</sup>

The quantity *a*Ξ(**x**,*t*) indicates how peaked the distribution *q*Ξ(**Ξ**(**x**,*t*)) is, i.e. how deterministic are the terms in (16) describing the medium: the larger *a*Ξ(**x**,*t*) is, the less stochastic is the

3 , , . *a t at t t qt e* 

<sup>3</sup> ,, ,

**BV BV B BV V**

, , 0 ,

*P d d W tt*

3

, [ , , , ; ,) , .

0

0 0 *W tt d d A tt* [ , ; ,) [ , , , ; , ) . **B V Ω Π ΩΠ B V** (21)

, [ , ; ,) . *t t t t*

**B BV V B V BV** (22)

0

**<sup>x</sup> <sup>x</sup> <sup>Ξ</sup> <sup>x</sup> <sup>Ξ</sup> <sup>x</sup> <sup>Ξ</sup> <sup>x</sup>** (23)

f f

*t t*

3

, ,,

**Ω Π B V**

*i d L dx*

terms will be collected in a noise-Lagrangian *LC*, so that all in all one has:

[ , , , ; ,) ,

*L L L A t t Nt t e*

**Ω Π B V**

0

*C tt e*

**Ω Π B V**

*C*

a kernel *W* involving only physical fields

should be calculated as

of graphs.

0

*t C t*

0 00

(17), each corresponding to a particular realization of **Ξ**, **Δ**, and **Θ** (Haken, 1983); then **B** and **V** can be arbitrarily irregular, because they inherit stochasticity from noises; they will possibly show sudden changes in time or non-differentiable behaviours in space, as it happens in ISCs. The description of such a system may be given in terms of *path integrals* (Feynman & Hibbs, 1965). The positions (16) and their consequence (17) are chosen because they reproduce exactly the Langevin equations treated in Phythian (1977), on which this model is based.

The construction introduced in the just mentioned work is the definition of a path integral scheme out of a suitable set of Langevin equations. One starts with a dynamical variable *ψ*, with any number of components, undergoing a certain equation with noises. Then, another variable *χ* is defined, referred to as *stochastic momentum conjugated to ψ*. In this way, it is possible to define a kernel

$$A\{\boldsymbol{\nu},\boldsymbol{\chi};\boldsymbol{t}\_{0},\boldsymbol{t}\} = N\left(\boldsymbol{t}\_{0},\boldsymbol{t}\right)e^{-i\int\_{0}^{\frac{\boldsymbol{t}}{\boldsymbol{\nu}}\cdot\mathbb{L}\left(\boldsymbol{\nu},\boldsymbol{\chi}\right)d\boldsymbol{\tau}}},\tag{18}$$

so that any statistical outcome of the history of the system between *t*0 and *t* is calculated as:

$$
\langle F \rangle = \iint d\boldsymbol{\psi} \, \overline{\boldsymbol{\Box}} \, \overline{\boldsymbol{\Box}} \, \overline{\boldsymbol{A}} \, \overline{\boldsymbol{A}} [\boldsymbol{\varphi}\_{\prime}, \boldsymbol{\chi}; \mathbf{t}\_{0}, \mathbf{t}] \boldsymbol{F}(\boldsymbol{\psi}) \, \boldsymbol{\Box}
$$

In the kernel in (18) the quantity *L*(*ψ*,*χ*) is referred to as *stochastic Lagrangian of the system*. In Phythian (1977) the key result is a closed "recipe" to build up *L*(*ψ*,*χ*) out of the Langevin equation of motion.

The same procedure may be applied to the system governed by the Langevin equations (17); these may be turned into a SFT by identifying the dynamical variables *ψ* of the system as **B** and **V**, and introducing as many stochastic momenta *χ* as the components of *ψ* (Materassi & Consolini, 2008):

$$
\boldsymbol{\nu} = \mathbf{B} \oplus \mathbf{V}, \quad \boldsymbol{\chi} = \mathbf{Q} \oplus \mathbf{II}.
$$

The variables **Ω** and **Π** are two vector quantities representing the stochastic momenta of **B** and **V** respectively. A stochastic kernel *A*[**Ω**,**Π**,**B**,**V**;*t*0,*t*) is constructed by involving a noise factor

$$\begin{aligned} \mathbf{C}[\boldsymbol{\Delta}, \boldsymbol{\Pi}, \mathbf{B}, \mathbf{V}; t\_0, t) &= \\ \mathbf{I} &= \iiint [d\boldsymbol{\Xi} \prod d\boldsymbol{\Delta} \, \mathbf{\overline{d}} \, d\boldsymbol{\Theta} \,] \mathbf{Q} [\boldsymbol{\Xi}\_{\boldsymbol{\Delta}} \boldsymbol{\Delta}, \mathbf{\Theta}] \, e^{i \int\_0^\cdot d\boldsymbol{\tau} \left[ \boldsymbol{d}^3 \, \mathbf{x} \left[ \boldsymbol{\Xi} \cdot \mathbf{\Omega} + \boldsymbol{\Theta} \cdot \boldsymbol{\Pi} + \boldsymbol{\Delta} \cdot (\mathbf{\Pi} \times \mathbf{\mathcal{B}}) \right] \right]} \, \end{aligned} \tag{19}$$

all the statistical dynamics of the resistive MHD interpreted as a SFT is then encoded in the kernel

$$\begin{split} A[\mathfrak{Q}, \Pi, \mathbf{D}, \mathbf{V}; t\_0, t) &= N(t\_0, t) \mathbf{C}[\mathfrak{Q}, \Pi, \mathbf{B}, \mathbf{V}; t\_0, t) e^{-i \int\_{t\_0}^{t} d\tau \left[ I\_0(\mathfrak{Q}, \Pi, \mathbf{B}, \mathbf{V}) \right] d^3 x} \\\\ L\_0(\mathfrak{Q}, \Pi, \mathbf{B}, \mathbf{V}) &= \mathbf{Q} \cdot \dot{\mathbf{B}} + \Pi \cdot \dot{\mathbf{V}} + \\ &+ \mathbf{Q} \cdot \left( \left( \mathbf{V} \cdot \hat{\boldsymbol{\varepsilon}} \right) \mathbf{B} - \left( \mathbf{B} \cdot \hat{\boldsymbol{\varepsilon}} \right) \mathbf{V} \right) + \Pi \cdot \left( \left( \mathbf{V} \cdot \hat{\boldsymbol{\varepsilon}} \right) \mathbf{V} \right) \,. \end{split} \tag{20}$$

46 Topics in Magnetohydrodynamics

(17), each corresponding to a particular realization of **Ξ**, **Δ**, and **Θ** (Haken, 1983); then **B** and **V** can be arbitrarily irregular, because they inherit stochasticity from noises; they will possibly show sudden changes in time or non-differentiable behaviours in space, as it happens in ISCs. The description of such a system may be given in terms of *path integrals* (Feynman & Hibbs, 1965). The positions (16) and their consequence (17) are chosen because they reproduce exactly

The construction introduced in the just mentioned work is the definition of a path integral scheme out of a suitable set of Langevin equations. One starts with a dynamical variable *ψ*, with any number of components, undergoing a certain equation with noises. Then, another variable *χ* is defined, referred to as *stochastic momentum conjugated to ψ*. In this way, it is

0 0 [ , ; ,) , ,

[ , ; ,) . <sup>0</sup>

so that any statistical outcome of the history of the system between *t*0 and *t* is calculated as:

In the kernel in (18) the quantity *L*(*ψ*,*χ*) is referred to as *stochastic Lagrangian of the system*. In Phythian (1977) the key result is a closed "recipe" to build up *L*(*ψ*,*χ*) out of the Langevin

The same procedure may be applied to the system governed by the Langevin equations (17); these may be turned into a SFT by identifying the dynamical variables *ψ* of the system as **B** and **V**, and introducing as many stochastic momenta *χ* as the components of *ψ* (Materassi &

> **B V** , . **Ω Π**

The variables **Ω** and **Π** are two vector quantities representing the stochastic momenta of **B** and **V** respectively. A stochastic kernel *A*[**Ω**,**Π**,**B**,**V**;*t*0,*t*) is constructed by involving a noise factor

all the statistical dynamics of the resistive MHD interpreted as a SFT is then encoded in the

[ , , , ; ,) , [ , , , ; ,) ,

00 0

0

*t t i d dx*

*F d d A t tF*

 

*A t t Nt te*

*d d dQ e*

**Ξ Δ Θ ΞΔΘ**

**Ω V BB V Π V V**

*A t t N t tC t te*

**Ω Π B V Ω Π B V**

**Ω Π B V Ω B Π V**

<sup>0</sup> [ , , , ; ,)

*C tt*

**Ω Π B V**

, ,,

0

*L*

  <sup>0</sup>

*t t iL d*

,

 

<sup>3</sup>

**ΞΩ ΘΠ Δ Π B**

0 0

, ,,

**Ω Π B V**

*idL d x*

(19)

3

*t t*

 

(20)

,, :

.

(18)

 

the Langevin equations treated in Phythian (1977), on which this model is based.

possible to define a kernel

equation of motion.

Consolini, 2008):

kernel

The quantity *L*0(**Ω**,**Π**,**B**,**V**) is interpreted as the part of the Lagrangian of the SFT not containing noise terms. *L*0 shows only space- and time-local terms, always: as it is stressed in Chang (1999), the integration of the noise term *C*[**Ω**,**Π**,**B**,**V**;*t*0,*t*) brings terms in *L* that are non-local in space and in time, due to the self- and mutual correlations of noises. Those terms will be collected in a noise-Lagrangian *LC*, so that all in all one has:

$$\begin{split} \mathbf{C}[\boldsymbol{\bf{\bf{\bf{E}},\boldsymbol{\bf{D}},\boldsymbol{\bf{D}},\boldsymbol{\bf{B}}},\mathbf{V};t\_{0},t) &= e^{-i\int\_{0}^{\frac{\hat{\boldsymbol{I}}}{\hat{\boldsymbol{I}}}d\boldsymbol{\bf{r}}\boldsymbol{f}\cdot\mathbf{L}\_{c}(\boldsymbol{\bf{Q}},\boldsymbol{\bf{D}},\mathbf{B},\mathbf{V})d^{3}\boldsymbol{x}}} \\ \boldsymbol{L} &= \boldsymbol{L}\_{\boldsymbol{C}} + \boldsymbol{L}\_{0}\boldsymbol{\iota} \quad A[\boldsymbol{\bf{Q}},\mathbf{I},\mathbf{I},\mathbf{B},\mathbf{V};t\_{0},t) = N\left(t\_{0},t\right)e^{-i\int\_{0}^{\frac{\hat{\boldsymbol{I}}}{\hat{\boldsymbol{I}}}d\boldsymbol{\pi}\boldsymbol{f}\cdot\mathbf{L}(\boldsymbol{\bf{B}},\mathbf{I},\mathbf{B},\mathbf{V})d^{3}\boldsymbol{x}}} \end{split}$$

The form of *Q*[**Ξ**,**Δ**,**Θ**], hence of *C*[**Ω**,**Π**,**B**,**V**;*t*0,*t*), may render the SFT long-range correlated and with a finite memory: these conditions of the ISC plasmas described by such a SFT is what encourages people to work through the techniques of *dynamical renormalization group* (Chang et al., 1978). Possibly, the stochastic momenta may be eliminated, so that one obtains a kernel *W* involving only physical fields

$$\mathcal{W}\{\mathbf{B},\mathbf{V};t\_0,t\} = \left[ \left[ \underline{d}\mathbf{2} \right] \left[ \left[ \underline{d}\mathbf{1} \right] A[\mathbf{Q},\mathbf{II},\mathbf{B},\mathbf{V};t\_0,t\right] \right. \tag{21}$$

Once *W*[**B**,**V**;*t*0,*t*) has been obtained, the calculation of processes in which the magnetized plasma changes arbitrarily, from an initial configuration (**B**(*t*0),**V**(*t*0)) = (**B**i,**V**i) to a final one (**B**(*t*),**V**(*t*)) = (**B**f,**V**f), may be done, for any time interval (*t*0,*t*): the rate of such transitions should be calculated as

$$P\_{\mathbf{B}\_{i},\mathbf{V}\_{i}\rightarrow\mathbf{B}\_{i},\mathbf{V}\_{i}} = \int [\![\![\![\mathbf{B}\!]\!]\!]\!] \left[\![\![\mathbf{B}\!]\!]\!] \mathcal{N}[\![\mathbf{B}\!]\!] \mathbf{V}\_{i}!t\_{0} \!] \right|\_{\mathbf{B}(t\_{0})=\mathbf{B}\_{i},\mathbf{V}(t\_{0})=\mathbf{V}\_{i}}.\tag{22}$$

As a further development of Materassi & Consolini (2008), a complete representation *à la Feynman* of such processes is to be derived from the SFT, with a suitable perturbative theory of graphs.

In order to arrive to a closed expression for a stochastic action at least in one example case, hereafter a toy model is reported, in which **Ξ**, **Δ** and **Θ** are assumed to be *Gaussian processes without any memory, and δ-correlated in space*. This hypothesis is surely over-simplifying for a plasma in ISC, since there are experimental results stating the presence of non-Gaussian distributions (Yordanova et al., 2005), and also of memory effects (Consolini et al., 2005). Nevertheless, the Gaussian example is of some use in illustrating the SFT at hand, because a Gaussian shape for *Q*[**Ξ**,**Δ**,**Θ**] allows for the full integration of *C*[**Ω**,**Π**,**B**,**V**;*t*0,*t*), and the explicit calculation of *W*[**B**,**V**;*t*0,*t*) from *A*[**Ω**,**Π**,**B**,**V**;*t*0,*t*) in (21). The probability density functional *Q*[**Ξ**,**Δ**,**Θ**] is obtained via a *continuous product* out of distributions of the local values of the fields **Ξ**, **Δ** and **Θ** of Gaussian nature; for instance, the PDF of the local variable **Ξ**(**x**,*t*) reads:

$$q\_{\Xi} \left( \Xi(\mathbf{x}, t) \right) = \sqrt{\frac{a\_{\Xi}^{3}(\mathbf{x}, t)}{\pi^{3}}} e^{-a\_{\Xi}(\mathbf{x}, t) \left( \Xi(\mathbf{x}, t) - \Xi\_{0}(\mathbf{x}, t) \right)^{2}}.\tag{23}$$

The quantity *a*Ξ(**x**,*t*) indicates how peaked the distribution *q*Ξ(**Ξ**(**x**,*t*)) is, i.e. how deterministic are the terms in (16) describing the medium: the larger *a*Ξ(**x**,*t*) is, the less stochastic is the

Sub-Fluid Models in Dissipative Magneto-Hydrodynamics 49

There is an apparent "necessity" of making the choice (16) in order to follow the scheme traced in Phythian (1977). It could be useful to extend the reasoning presented here to other forms of the Langevin equations so to avoid the positions (16) and work directly with *ζ* and

It is also to mention that the problem of defining a good functional measure is still to be examined, by studying the consistency condition of a Fokker-Planck equation for the SFT, starting for example with the Lagrangian density (25), obtained under drastically

A comment is deserved by the choice of the *incompressible plasma hypothesis*. The MHD as a dynamical system is given by (5): in the absence of incompressibility, the mass density *ρ* is a distinct variable on its own, with a proper independent dynamics. In the stochastic theory *à la Phythian* each dynamical variable should satisfy a Langevin equation, in which noise is in principle involved. Now, altering the equation for *ρ* with noise could invalidate the mass conservation, which is a big fact one would like to avoid. Hence, the "sacred principle" of non-relativistic mass conservation ∂*tρ* + ∂·(*ρ***V**) = 0 is saved excluding *ρ* from dynamics, rendering it a pure parameter of the theory, via incompressibility. The compressible case could be studied considering the local mass conservation a constrain to be imposed to the path integrals as it happens in quantum gauge field theories (Hennaux & Teitelboim, 1992). Last but not least, the fourth equation in (5) has not been considered at all in this scheme: in Phythian's scheme plasma thermodynamics must be discussed in some deeper way before

Among the many interesting fast and irreversible processes occurring in plasmas, *magnetic reconnection* is surely one of the most important (see e.g. Biskamp, 2000; Birn and Priest, 2007). The name "magnetic reconnection", originally introduced by Dungey (1953), refers to a process in which a particle acceleration is observed consequently to a change of the magnetic field line topology (*connectivity*). Being associated to a change in the magnetic field line topology, the magnetic reconnection process involves the occurrence of magnetic field line diffusion, disconnection and reconnection and it is also accompanied by plasma heating and particle acceleration, sometimes termed as *dissipation* (actually, in this case dissipation means transfer of energy from the magnetic field to the particle energy, both bulk motion energy, the term *ρV*2/2 in the integrand in (7), and thermal energy, the term *U*(*ρ*,*s*) in the same expression of *H*; in the context of metriplectic dynamics, dissipation is simply the

The traditional approach to magnetic reconnection is based on resistive MHD theory. In this framework one of the most famous and first scenarios of magnetic reconnection, able to make some quantitative predictions, was proposed by Parker (1957) and Sweet (1958). The Sweet-Parker model provides a simple 2-dimensional description of steady magnetic reconnection in a non-compressible plasmas (see Figure 1). In this model there are two relevant scales: the global scale *L* of the magnetic field and the thickness Δ of the current sheet (or of the diffusion region). The main result of such a model may be resumed in the

enlarging the configuration space of stochastic fields to the entropy *s*.

**4. Fractal model of fast reconnection** 

transfer of energy into the addendum *U*(*ρ*,*s*)).

very-well known expression for the *Alfvèn Mach number M*A,

*p* as stirring forces in (14).

simplifying hypotheses.

plasma. Formally equal distributions *q*Δ(**Δ**(**x**,*t*)) and *q*Θ(**Θ**(**x**,*t*)) describe the local occurrence of the values of **Δ** and **Θ**. From (23), the expression of the noise kernel *C*[**Ω**,**Π**,**B**,**V**;*t*0,*t*) defined in (19) can be calculated explicitly (Materassi & Consolini, 2008), and the noise Lagrangian *LC*(**Ω**,**Π**,**B**,**V**) determined in a closed form:

$$\begin{split} L\_{\mathbb{C}} \left( \mathbf{\tilde{\mathbf{\mathbf{Q}}}}, \Pi, \mathbf{\tilde{\mathbf{B}}}, \mathbf{V} \right) &= -\frac{i\mathbf{\tilde{\mathbf{Q}}^{2}}}{4a\_{\Xi}} - \Xi\_{0} \cdot \mathbf{\mathcal{Q}} + \\ -\frac{i}{4} \left( \frac{\Pi^{2}}{a\_{\Theta}} + \frac{\Pi^{2}\mathbf{B}^{2} - \left(\mathbf{B} \cdot \Pi\right)^{2}}{a\_{\Delta}} \right) - \left( \Theta\_{0} \cdot \Pi + \left(\mathbf{B} \times \Delta\right) \cdot \Pi \right) \end{split} \tag{24}$$

This noise Lagrangian is space-local and does not contain any memory term, because the PDF *Q*[**Ξ**,**Δ**,**Θ**] was constructed as the continuous products of infinite terms, each of which representing the independent probability *q*Ξ(**Ξ**(**x**,*t*))*q*Δ(**Δ**(**x**,*t*))*q*Θ(**Θ**(**x**,*t*)). The total Lagrangian is the sum of the noise term *LC*(**Ω**,**Π**,**B**,**V**) and of the "deterministic" addendum *L*0(**Ω**,**Π**,**B**,**V**) presented in (20). The sum *L*0 + *LC* gives rise to a perfectly local theory. The total Lagrangian *L*0 + *LC* gives a kernel *A*[**Ω**,**Π**,**B**,**V**;*t*0,*t*) that is the continuous product of the exponentiation of quadratic terms in **Ω** and **Π**, so that the calculation (21) is an infinite-dimensional Gaussian path integral, which is again feasible. This means that, under the hypothesis (23) on **Ξ**, and similar assumptions on the two other noises **Δ** and **Θ**, the calculation of the stochastic evolution kernel can be done in terms of pure "physical fields" **B** and **V**, obtaining *W*[**B**,**V**;*t*0,*t*). If the calculation is performed to the end, the expression of *W*[**B**,**V**;*t*0,*t*) reads:

 3 0 0 0 2 0 0 2 0 0 ' , 0 0 00 2 2 2 1 2 0 2 1 2 0 2 [ , ; ,) ' , , , , ; , , ' , ln 1 1 . *t t i d L dx a a a a a a S tt Na a a ptte L i ia ia p p* **B V B B B B B B B V B V B BV BB V B VV V B VV V B** (25)

The functions *ζ*0 and *p*0 are defined as the ensemble expectation value of the homonymous stochastic variables. The expression (25) is ready to be used in (22) to calculate the transition probabilities between arbitrary field configurations. The quantity *N'* in (25), whatever it looks like, will not enter the calculations of processes like (22), since it doesn't depend on **V** and B, and will be cancelled out. Last but not least, consider that the functions defining noise statistics, i.e. *a*Ξ, *a*Δ, *a*Θ, *ζ*0 and *p*0, do enter the Lagrangian as "coupling constants".

*Intrinsic limitations* of the proposed scheme can be recognized.

First of all, no discussion has been even initiated yet about the convergence of all the quantities defined.

48 Topics in Magnetohydrodynamics

plasma. Formally equal distributions *q*Δ(**Δ**(**x**,*t*)) and *q*Θ(**Θ**(**x**,*t*)) describe the local occurrence of the values of **Δ** and **Θ**. From (23), the expression of the noise kernel *C*[**Ω**,**Π**,**B**,**V**;*t*0,*t*) defined in (19) can be calculated explicitly (Materassi & Consolini, 2008), and the noise

0

2

This noise Lagrangian is space-local and does not contain any memory term, because the PDF *Q*[**Ξ**,**Δ**,**Θ**] was constructed as the continuous products of infinite terms, each of which representing the independent probability *q*Ξ(**Ξ**(**x**,*t*))*q*Δ(**Δ**(**x**,*t*))*q*Θ(**Θ**(**x**,*t*)). The total Lagrangian is the sum of the noise term *LC*(**Ω**,**Π**,**B**,**V**) and of the "deterministic" addendum *L*0(**Ω**,**Π**,**B**,**V**) presented in (20). The sum *L*0 + *LC* gives rise to a perfectly local theory. The total Lagrangian *L*0 + *LC* gives a kernel *A*[**Ω**,**Π**,**B**,**V**;*t*0,*t*) that is the continuous product of the exponentiation of quadratic terms in **Ω** and **Π**, so that the calculation (21) is an infinite-dimensional Gaussian path integral, which is again feasible. This means that, under the hypothesis (23) on **Ξ**, and similar assumptions on the two other noises **Δ** and **Θ**, the calculation of the stochastic evolution kernel can be done in terms of pure "physical fields" **B** and **V**, obtaining *W*[**B**,**V**;*t*0,*t*). If the calculation is performed to the end, the expression of *W*[**B**,**V**;*t*0,*t*) reads:

*a*

2 2 2 2

**<sup>Ω</sup> Ω Π B V <sup>Ω</sup>**

**Π Π B B Π**

**Θ Π B Δ Π**

' ,

**B V**

*i d L dx*

2

.

0

*t t*

2

**B B B**

 

2 0 0

  3

. (24)

(25)

0

Lagrangian *LC*(**Ω**,**Π**,**B**,**V**) determined in a closed form:

4

*i*

*C*

*<sup>i</sup> <sup>L</sup>*

, ,, <sup>4</sup>

*a a*

**B V B**

' , ln 1

*L i*

1

*ia*

**B V**

*a a*

quantities defined.

  *a a*

 

*Intrinsic limitations* of the proposed scheme can be recognized.

**B**

*ia*

**BV BB V B**

2 0 2

2

[ , ; ,) ' , , , , ; , ,

0 0 00

*a a*

**VV V**

 

*S tt Na a a ptte*

 *p*

**VV V B**

 

*p*

1

0 2

The functions *ζ*0 and *p*0 are defined as the ensemble expectation value of the homonymous stochastic variables. The expression (25) is ready to be used in (22) to calculate the transition probabilities between arbitrary field configurations. The quantity *N'* in (25), whatever it looks like, will not enter the calculations of processes like (22), since it doesn't depend on **V** and B, and will be cancelled out. Last but not least, consider that the functions defining noise statistics, i.e. *a*Ξ, *a*Δ, *a*Θ, *ζ*0 and *p*0, do enter the Lagrangian as "coupling constants".

First of all, no discussion has been even initiated yet about the convergence of all the

2 0 0

1 2

**B B B**

0 0

There is an apparent "necessity" of making the choice (16) in order to follow the scheme traced in Phythian (1977). It could be useful to extend the reasoning presented here to other forms of the Langevin equations so to avoid the positions (16) and work directly with *ζ* and *p* as stirring forces in (14).

It is also to mention that the problem of defining a good functional measure is still to be examined, by studying the consistency condition of a Fokker-Planck equation for the SFT, starting for example with the Lagrangian density (25), obtained under drastically simplifying hypotheses.

A comment is deserved by the choice of the *incompressible plasma hypothesis*. The MHD as a dynamical system is given by (5): in the absence of incompressibility, the mass density *ρ* is a distinct variable on its own, with a proper independent dynamics. In the stochastic theory *à la Phythian* each dynamical variable should satisfy a Langevin equation, in which noise is in principle involved. Now, altering the equation for *ρ* with noise could invalidate the mass conservation, which is a big fact one would like to avoid. Hence, the "sacred principle" of non-relativistic mass conservation ∂*tρ* + ∂·(*ρ***V**) = 0 is saved excluding *ρ* from dynamics, rendering it a pure parameter of the theory, via incompressibility. The compressible case could be studied considering the local mass conservation a constrain to be imposed to the path integrals as it happens in quantum gauge field theories (Hennaux & Teitelboim, 1992).

Last but not least, the fourth equation in (5) has not been considered at all in this scheme: in Phythian's scheme plasma thermodynamics must be discussed in some deeper way before enlarging the configuration space of stochastic fields to the entropy *s*.
