Single-particle velocity distributions of collisionless, steady-state plasmas must follow Superstatistics

TL;DR

This paper develops a statistical mechanical framework for collisionless, steady-state plasmas that derives velocity distributions without entropy, recovering superstatistics as a special case and introducing a microscopic inverse temperature definition.

Contribution

It introduces a probability-based formalism for plasma velocity distributions, generalizing superstatistics and avoiding reliance on entropy concepts.

Findings

Superstatistics is recovered as a limiting case.

A microscopic definition of inverse temperature is provided.

Non-extensivity arises from correlations, not assumed a priori.

Abstract

The correct modelling of velocity distribution functions for particles in steady-state plasmas is a central element in the study of nuclear fusion and also in the description of space plasmas. In this work, a statistical mechanical formalism for the description of collisionless plasmas in a steady state is presented, based solely on the application of the rules of probability and not relying on the concept of entropy. Beck and Cohen's superstatistical framework is recovered as a limiting case, and a "microscopic" definition of inverse temperature $\beta$ is given. Non-extensivity is not invoked a priori but enters the picture only through the analysis of correlations between parts of the system.

Equations

$$L=\sum _{i=1}^N\left(-\frac{m_i{c}^2}{\gamma ({\mathbf{v}}_i)}-q_{i}V({\mathbf{r}}_i;t)+q_i{\mathbf{v}}_i\cdot \mathbf{A}({\mathbf{r}}_i;t)\right),$$

$$\gamma (\mathbf{v})=1/\sqrt{1-(v/c{)}^2}.$$

$${\mathbf{p}}_i=\frac{\partial L}{\partial {\mathbf{v}}_i}=\gamma ({\mathbf{v}}_i)m_i{\mathbf{v}}_{\mathbf{i}}+q_i\mathbf{A}({\mathbf{r}}_i;t),$$

$$E(\mathbf{R},\mathbf{V};t)$$

$$=\sum _{i=1}^N{K}_i({\mathbf{v}}_i)+\Phi (\mathbf{R};t),$$

$$\left[\frac{\partial}{\partial t}+\sum_{i=1}^{N}\Big{(}\bm{v}_{i}\cdot\frac{\partial}{\partial\bm{r}_{i}}+\bm{a}_{i}\cdot\frac{\partial}{\partial\bm{v}_{i}}\Big{)}\right]P(\bm{R},\bm{V}|t)=0,$$

$$\frac{d}{dt}\left(\gamma ({\mathbf{v}}_i)m_i{\mathbf{v}}_i\right)=q_i\left(\mathbf{E}({\mathbf{r}}_i;t)+{\mathbf{v}}_i\times \mathbf{B}({\mathbf{r}}_i;t)\right).$$

$$\nabla \cdot \mathbf{E}$$

$$\nabla \times \mathbf{E}$$

$$\nabla \cdot \mathbf{B}$$

$$\nabla \times \mathbf{B}$$

$$\rho (\mathbf{r};t)$$

$$=\sum _{i=1}^N{q}_i\times \int d\mathbf{R}d\mathbf{V}P(\mathbf{R},\mathbf{V}|t)\delta ({\mathbf{r}}_i-\mathbf{r}),$$

$$\mathbf{J}(\mathbf{r};t)$$

$$=\sum _{i=1}^N{q}_i\times \int d\mathbf{R}d\mathbf{V}P(\mathbf{R},\mathbf{V}|t){\mathbf{v}}_i\delta ({\mathbf{r}}_i-\mathbf{r}).$$

$$\frac{dE}{dt}=\frac{\partial E}{\partial t}+\sum _{i=1}^N\left({\mathbf{v}}_i\cdot \frac{\partial E}{\partial {\mathbf{r}}_i}+{\mathbf{a}}_i\cdot \frac{\partial E}{\partial {\mathbf{v}}_i}\right)=0.$$

$$\sum_{i=1}^{N}\Big{(}\bm{v}_{i}\cdot\frac{\partial}{\partial\bm{r}_{i}}+\bm{a}_{i}\cdot\frac{\partial}{\partial\bm{v}_{i}}\Big{)}P(\bm{R},\bm{V}|\mathcal{S})=0.$$

$$P(\mathbf{R},\mathbf{V}|\mathcal{S})=p(E(\mathbf{R},\mathbf{V})).$$

$$P(E|\mathcal{S})=p(E)\Omega (E),$$

$$p(E)={\int }_0^{\infty }d\beta f(\beta )\exp (-\beta E),$$

$$P(\mathbf{\Gamma }|\mathcal{S})$$

$$={\int }_0^{\infty }d\beta P(\beta |\mathcal{S})\left[\frac{\exp (-\beta H(\mathbf{\Gamma }))}{Z(\beta )}\right].$$

$$P(\beta |\mathcal{S})=f(\beta )Z(\beta ).$$

$${\beta }_F(E):=-\frac{d}{dE}\ln p(E),$$

$$\Big{<}\nabla\cdot\bm{\omega}\Big{>}_{\beta}=\beta\Big{<}\bm{\omega}\cdot\nabla H\Big{>}_{\beta}.$$

$$\displaystyle\Big{<}\nabla\cdot\bm{\omega}\Big{>}_{\mathcal{S}}$$

$$\displaystyle=\Big{<}\beta\Big{<}\bm{\omega}\cdot\nabla H\Big{>}_{\beta}\Big{>}_{\mathcal{S}}$$

$$\displaystyle=\Big{<}\beta\bm{\omega}\cdot\nabla H\Big{>}_{\mathcal{S}}.$$

$$\displaystyle\Big{<}\nabla\cdot\bm{\omega}\Big{>}_{\mathcal{S}}$$

$$\displaystyle=\Big{<}\beta_{F}(H)\bm{\omega}\cdot\nabla H\Big{>}_{\mathcal{S}}.$$

$$\Big{<}\beta\bm{\omega}\cdot\nabla H\Big{>}_{\mathcal{S}}=\Big{<}\beta_{F}(H)\bm{\omega}\cdot\nabla H\Big{>}_{\mathcal{S}},$$

$$\Big{<}\beta G(\bm{\Gamma})\Big{>}_{\mathcal{S}}=\Big{<}\beta_{F}(H)G(\bm{\Gamma})\Big{>}_{\mathcal{S}}$$

$$\beta_{F}(E)=\frac{\Big{<}\beta\delta(E-H(\bm{\Gamma}))\Big{>}_{\mathcal{S}}}{\Big{<}\delta(E-H(\bm{\Gamma}))\Big{>}_{\mathcal{S}}}=\Big{<}\beta\Big{>}_{\mathcal{S},H=E}.$$

$$\Big{<}\beta_{F}\Big{>}_{\mathcal{S}}=\Big{<}\beta\Big{>}_{\mathcal{S}}.$$

$$\tilde{E}=E-k_1=\Phi +\sum _{i=2}^N{k}_i.$$

Full text

I Introduction

Plasmas in nonequilibrium steady states sometimes follow non-Maxwellian velocity distributions, one of the most common known as Kappa distributions Lima et al. (2000); Leubner (2002); Livadiotis (2017). Properties of Kappa-distributed plasmas have been extensively studied Gougam and Tribeche (2011); Viñas et al. (2015); Livadiotis (2014, 2016), however their origin is still a matter of debate.

One of the main arguments put forward for the existence of Kappa distributions in plasmas relies on the non-extensive statistics proposed by Constantino Tsallis Tsallis (2009), which postulates a generalization of the Boltzmann-Gibbs entropy depending on a non-extensive index $q$. More recently, the formalism known as superstatistics Beck (2004) has been gaining attention, as it is able to produce Kappa distributions without extending the entropy: it asserts that non-Maxwellian distributions arise from a superposition of canonical ensembles at different values of the inverse temperature $\beta=1/k_{B}T$. Interestingly, superstatistics as a tool to explain the non-Maxwellian distributions found in plasmas has only recently been properly acknowledged in the literature Ourabah et al. (2015).

In this work we present a first-principles derivation of the single-particle energy distribution for steady-state plasmas, using only the tools of standard statistical mechanics. We show that, in general, the velocity distribution of a single particle follows the superstatistical formalism and we give an interpretation of the superstatistical (inverse) temperature distribution in this case. This work opens the way for a direct test of the superstatistical formalism against explicit kinetic simulations, for instance, based on particle-in-cell methods.

II Statistical description of plasma

As it turns out, in most situations only the high-energy tails of the velocity distribution in plasmas of interest (be it laboratory or space plasmas) reach relativistic speeds. However, without any additional effort we can obtain the fully relativistic results and then take the low energy limit for non-relativistic cases. Hence we will start with a fully relativistic statistical mechanical description of $N$ charged particles subjected to fixed, time-dependent electromagnetic fields. The Lagrangian of such a system is given by

[TABLE]

where $V(\bm{r};t)$ and $\bm{A}(\bm{r};t)$ are the scalar and vector potentials, respectively, and $\gamma(\bm{v})$ is the Lorentz factor,

[TABLE]

The canonical momentum $\bm{p}_{i}$ associated to $\bm{r}_{i}$ is given by

[TABLE]

and the Hamiltonian is $H=\sum_{i=1}^{N}(\bm{p}_{i}\cdot\bm{v}_{i})-L$. Let us introduce the $3N$-component vectors $\bm{R}$ and $\bm{V}$ as notation shortcuts for $(\bm{r}_{1},\ldots,\bm{r}_{N})$ and $(\bm{v}_{1},\ldots,\bm{v}_{N})$ respectively. Instead of working in the canonical variables $\bm{r}_{i}$ and $\bm{p}_{i}$ it is convenient to go back to positions and velocities, as in this way we can write an energy function $E(\bm{R},\bm{V};t)$, which is nothing but $H(\bm{R},\bm{P};t)$ rewritten to eliminate $\bm{p}_{i}$ using Eq. 2,

[TABLE]

and which is independent of $\bm{A}$. We see that $E$ is simply the sum of kinetic and potential energies, as the terms involving $q_{i}\bm{v}_{i}\cdot\bm{A}(\bm{r}_{i};t)$ have cancelled out.

The description of this system in terms of statistical mechanics in absence of collisions is given by the Liouville equation, where we have taken the 6$N$-dimensional vectors $(\bm{R},\bm{V})$ as the available microstates of the system. The Liouville equation can written in our case in the form Ichimaru (1992),

[TABLE]

where $P=P(\bm{R},\bm{V}|t)$ is the time-dependent probability density of microstates, and $\bm{a}_{i}=d\bm{v}_{i}/dt$ is the acceleration of each particle, which is determined by the electric and magnetic fields through the Lorenz force,

[TABLE]

In the case of plasma, the potentials $V$ and $\bm{A}$ (and therefore the fields $\bm{E}$ and $\bm{B}$) include contributions from interactions between the charged particles themselves, as well as the external fields. Then in practice, the Liouville equation is self-consistently solved together with Maxwell’s equations,

[TABLE]

where $\rho$ and $\bm{J}$ are the charge and current densities, respectively, which are functionals of the time-dependent probability density of microstates $P(\bm{R},\bm{V}|t)$. The charge density is given explicitly in terms of $P(\bm{R},\bm{V}|t)$ by

[TABLE]

while the current density is connected to $P(\bm{R},\bm{V}|t)$ by

[TABLE]

The fact that the Liouville equation needs to be solved self-consistently with the Maxwell equations does not break any of our assumptions. This is because, for a given dynamical process of interest, if there is a self-consistent solution $V^{*}(\bm{r};t)$ and $\bm{A}^{*}(\bm{r};t)$ of Eq. 4 and II, then the system of particles can be described by the Lagrangian in Eq. 1 and the energy in Eq. 3 under those fixed, but time-dependent potentials.

In a steady state, by definition the microstate distribution $P(\bm{R},\bm{V}|t)$ does not depend on $t$, therefore $\rho$ and $\bm{J}$ are time-independent, according to Eqs. 7 and Eq. 8. Because of Maxwell’s equations, $V^{*}$ and $\bm{A}^{*}$ must also be time-independent. This implies that $E(\bm{R},\bm{V};t)=E(\bm{R},\bm{V})$ and $\partial E/\partial t=0$, so we have

[TABLE]

The steady-state solutions of the Liouville equation (Eq. 4), denoted as $P(\bm{R},\bm{V}|\mathcal{S})$, are such that

[TABLE]

Because of Jeans’ theorem Dendy (1995), $P(\bm{R},\bm{V}|\mathcal{S})$ must depend on $\bm{R}$ and $\bm{V}$ only through the integrals of motion. However, in order to be able to define a temperature, the solution of Eq. 10 can only depend on $(\bm{R},\bm{V})$ through the energy, that is,

[TABLE]

In the following, we will refer to $p(E)$ as the ensemble function It is important to make the distinction that $p(E)$, although a non-negative quantity, is not the probability distribution of the energy in the ensemble $\mathcal{S}$, which we will denote by $P(E|\mathcal{S})$ and is given by

[TABLE]

with $\Omega(\varepsilon)=\int d\bm{R}d\bm{V}\delta(\varepsilon-E(\bm{R},\bm{V}))$ the density of states.

In the next section we will consider the notion of temperature for systems in this kind of steady state.

III Superstatistics

The following discussion will be framed in general terms, for any system with microstates $\bm{\Gamma}$ in an ensemble described by $P(\bm{\Gamma}|\mathcal{S})=p(H(\bm{\Gamma}))$, although we have in mind of course the collisionless plasma described in the previous section, with $\bm{\Gamma}=(\bm{R},\bm{V})$ and $H(\bm{\Gamma})=E(\bm{R},\bm{V})$.

The formalism known as superstatistics, introduced by Beck and Cohen in 2003 Beck and Cohen (2003), considers a family of ensemble functions $p(E)$ of the form

[TABLE]

where $f(\beta)$ is a non-negative weight function. From a Bayesian point of view Sattin (2006), the correct way of writing the superstatistical ensemble is through the marginalization of $\beta$ from the joint distribution $P(\bm{\Gamma},\beta|\mathcal{S})$, that is,

[TABLE]

This form of superstatistics, where the partition function appears explicitly, is called type-B superstatistics, in order to distinguish it from the form in Eq. 13, which is then called type-A superstatistics. Here we define the proper probability distribution of $\beta$ as

[TABLE]

Although this probability distribution for $\beta$ is commonly interpreted in terms of fluctuations of a phase-space observable $\hat{\beta}(\bm{\Gamma})$, this additional assumption can lead to inconsistencies Davis and Gutiérrez (2018) and is not strictly needed by the theory.

IV The fundamental temperature

In the case of a steady-state ensemble, as defined in Eq. 11, it is possible to define an ensemble-dependent temperature function

[TABLE]

which we will call the fundamental inverse temperature. The motivation for using this quantity can be understood starting from the canonical equipartition theorem written in the form

[TABLE]

Now consider $\beta$ as a random variable under a new superstatistical ensemble $\mathcal{S}$, taking expectation on both sides over the joint distribution $P(\bm{\Gamma},\beta|\mathcal{S})$. Because \beta\big{<}A\big{>}_{\beta}=\big{<}\beta A\big{>}_{\beta} for any quantity $A$ we obtain

[TABLE]

On the other hand, applying the conjugate variables theorem Davis and Gutiérrez (2012) to the distribution $P(\bm{\Gamma}|\mathcal{S})=p(H(\bm{\Gamma}))$ and using the chain rule for the gradient of $\ln P(\bm{\Gamma}|\mathcal{S})$, we see that Palma et al. (2016)

[TABLE]

As both Eqs. 17 and 18 must be valid for any choice of $\bm{\omega}$, we find that

[TABLE]

hence

[TABLE]

for any function $G(\bm{\Gamma})$. With the particular choice $G(\bm{\Gamma})=\delta(E-H(\bm{\Gamma}))$ we see that

[TABLE]

In other words, the fundamental (inverse) temperature function at an energy $E$ is the mean superstatistical (inverse) temperature for all states at energy $E$. It also follows from Eq. 20 with $G=1$ that

[TABLE]

V The single-particle fundamental temperature

We are now ready to approach the problem of determining the possible velocity distributions of a particle in a steady-state plasma. At this point then, we will separate the energy of the system into two contributions, the kinetic energy $k_{1}$ of a single particle (let us say particle 1 without loss of generality) and the energy $\tilde{E}$ of its environment, so that

[TABLE]

Integration of Eq. 11 over the positions $\bm{R}$ and the velocities $\bm{v}_{2},\ldots,\bm{v}_{N}$ gives the probability density of $\bm{v}_{1}$ as

[TABLE]

where we have defined the density of states of the environment $\tilde{E}$ as

[TABLE]

Eq. 22 gives us the connection between the ensemble function $p(E)$ describing the full distribution of microstates and the ensemble function $p_{1}(k)$ that describes the velocity distribution of a single particle. It shows that $\bm{v}_{1}$ follows a distribution with ensemble function

[TABLE]

and then the probability of observing a kinetic energy $k_{1}$ in the ensemble $\mathcal{S}$ is given simply by

[TABLE]

in agreement with Eq. 12. Similarly, the joint distribution of energies of the subsystem $k$ and the environment $\tilde{E}$ is given by

[TABLE]

From the ensemble function $p_{1}(k)$ in Eq. 24 we will compute directly the single-particle fundamental temperature as

[TABLE]

which is the expectation of $\beta_{F}(E)$ (inside curly braces) under a particular distribution for $E$ (inside square brackets). Now we use the product rule of probability theory Jaynes (2003) to show that

[TABLE]

Replacing in Eq. 27 we can finally obtain the single-particle fundamental (inverse) temperature as

[TABLE]

This is an important result. It shows that the fundamental (inverse) temperature for the velocity of a single particle at energy $k$ is given by the conditional expectation of the fundamental (inverse) temperature for the complete system, when the kinetic energy $K_{1}(\bm{v}_{1})$ of the particle is fixed at $k$.

Here we can distinguish two cases:

1. (a)

If the effect of fixing the kinetic energy $K_{1}$ on the whole system is negligible, then $\beta^{(1)}_{F}(k)$ will be a constant $\beta_{0}$, independent of $k$, and the particle will then be described by the canonical ensemble

[TABLE]

at inverse temperature \beta_{0}=\big{<}\beta_{F}(E)\big{>}_{\mathcal{S}}. Note that because $K_{1}(\bm{v}_{1})=\gamma(v_{1})m_{1}c^{2}$, this coincides with the Jüttner distribution Nakamura (2009) for relativistic speeds, and reduces to the Maxwell-Boltzmann distribution for non-relativistic particles. 2. (b)

If the particles in the system are highly correlated (as in the case of plasma), then fixing $K_{1}=k$ does influence the states of the system, and therefore

[TABLE]

In this case, $\beta^{(1)}_{F}(k)$ will depend on $k$, producing a non-canonical ensemble.

In the next section we will show the conditions under which $P(\bm{v}_{1}|\mathcal{S})$ of Eq. 22 can be described by Superstatistics, and obtain an explicit expression for the superstatistical (inverse) temperature distribution $P(\beta|\mathcal{S})$.

VI Derivation of the superstatistical formalism

In order to simplify the description of the single-particle velocity distribution given by Eqs. 22 and 24, we will make the reasonable assumption that the energy $\tilde{E}$ of the environment is much larger than the kinetic energy of the particle, $\tilde{E}\gg k$. This is to be expected in plasmas because then the kinetic energies dominate over the potential energy Bellan (2006), so $\Phi$ will not be able to cancel $K^{\prime}=K-k_{1}$, even when the system is bound ($\Phi<0$). Under this assumption we can use the definition of $\beta_{F}(E)$ to approximate $p(\tilde{E}+k)$ in Eq. 24 as

[TABLE]

and hence obtain

[TABLE]

Now, in the case when $\beta_{F}(E)>0$ we can introduce the function

[TABLE]

and use it to rewrite $p_{1}(k)$ as

[TABLE]

This is precisely type-A superstatistics, while the corresponding expression in type-B superstatistics is obtained by including the single-particle partition function $Z_{1}$ as

[TABLE]

where

[TABLE]

and $\mathcal{K}_{2}(x)$ is the modified Bessel function of the second kind Greiner et al. (2012). We can give an explicit expression for $P(\beta|\mathcal{S})$ as

[TABLE]

where in the last equality we have used Eq. 26, together with the marginalization rule as

[TABLE]

Eq. 36 then tells us that $P(\beta|\mathcal{S})\approx P(\beta_{F}(\tilde{E})=\beta|\mathcal{S})$ and therefore

[TABLE]

This approximation allows us to understand the fluctuating temperature of the particle not in terms of its kinetic energy, but as the fundamental temperature of its environment. A direct consequence of this is the fact that a non-fluctuating environment will always induce an equilibrium distribution for $\bm{v}_{1}$, because then

[TABLE]

where $E_{0}$ is the constant energy of the environment, and in this case superstatistics (Eq. 14) reduces to the canonical ensemble.

VII Concluding Remarks

In summary, we have shown that it is possible to determine the velocity distribution function of a particle in a plasma given the original ensemble function of the complete system (or, equivalently, its fundamental temperature function), and the probability distribution of the environment energy $\tilde{E}$. This distribution belongs to the superstatistics family, and its parameter $\beta$ follows the same distribution as $\beta_{F}(\tilde{E})$.

Our interpretation of the superstatistical $\beta$ for the single particle velocity distribution is then connected to the energy fluctuations of the environment (and not of the system itself), and in absence of them, the particle follows a canonical (Jüttner or Maxwell-Boltzmann) distribution. If the particle is highly correlated with its environment (as is the case with plasmas), then in general the particle velocities are non-Maxwellian, because the fundamental temperature will be energy-dependent, as per Eq. 31. In particular, it follows from Eq. 37 that a sufficient condition for Kappa/Tsallis statistics is for $\beta_{F}(\tilde{E})$ to follow a gamma distribution.

VIII Acknowledgements

This work was supported by FONDECYT 1171127 and Anillo ACT-172101 grants.