Lectures on Non-Linear Plasma Kinetics (Springer Series on Atoms + Plasmas, 17)

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From the Wigner function, we can readily derive the marginal probability distributions in coordinate and momentum space. Indeed, from 2. For simplicity, assume all particles to have the same mass m. In analogy with 2. For simplicity, at this stage we are not taking into account the usually fermionic character of the charged particles in plasma, so that the N-body ensemble wavefunctions are not necessarily antisymmetrized.

In other words, f x1 , v1 ,.

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However, the Wigner function is not positive definite, so that it is not a probability but a pseudo-probability distribution. More exactly, suppose a classical phase-space function A x1 , v1 ,. We prefer to use velocities instead of momenta to assure a manifestly gauge invariant formalism. In addition, the transition from functions to operators is by no means unique: a well-defined correspondence rule should be employed [18, 26].

Special care should be paid with more complicated phase space observables involving noncommuting objects like products of functions of position and momenta. Indeed, to calculate expectation values using the Wigner formalism, we need first to map the observable into a phase-space function using the Weyl correspondence [37]. Given a classical function A x1 , v1 ,. As an example, one can be interested in the expectation value of the total energy of an interacting system with potential energy V x1 ,.

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Indeed, after some integrations by parts the right-hand side of 2. In addition to the Wigner function, alternative quantum probability distribution functions can be constructed. Among them, we can cite at least the Glauber— Sudarshan function [11, 34], the Q-function [12], the Husimi function [19], the Kirkwood distribution function [22] and the standard-ordered distribution function [32].

For these alternative functions, the underlying quantum-classical correspondence is given by specific methods other than the Weyl rule [8, 26]. The relevance of the non-Wigner probability distributions is recognized for specific purposes. For instance, the Q-functions and the Husimi functions can be shown to be everywhere nonnegative [26]. However, the Wigner function has a number of simultaneous attractive properties which makes it more popular than the other distribution functions.

For example, the Q and Husimi functions does not provide the marginal probability distributions in coordinate and momentum space in the sense of 2.

However, although their expressions are the same, their correspondence rules are not, see [26]. The other partial, or reduced Wigner functions would have a similar interpretation. What is the evolution equation satisfied by the N-body Wigner function? To answer the question, we follow the philosophy of [24]. From 2. However, the development of analytic and numerical techniques for the N-body problem is of course a tremendous task.

Moreover, for practical reasons, it is more effective to deal with the reduced Wigner functions, since f N contain far more information than what is needed.

We also observe that 2. We are specially concerned with the case where the system components interact through some two-body potential W , V x1 ,. Integrating 2. In the derivation N 1 was taken into account. Hence, in both the classical infinite BBGKY set of equations and its quantum analogue, we are faced with a closure problem.

The simpler way to deal with the truncation problem is by ignoring correlations, assuming that the distribution of particles at xi , vi is not affected by particles at a distinct phase space point x j , v j. For instance, such a circumstance arises in solid state devices, when considering the electronic motion in a fixed ionic lattice or under a confining field like in quantum wires or quantum wells [10, 20, 30].

Or even we can simply incorporate the field due to an homogeneous ionic background. In these cases, the external potential is of the form N Vext x1 ,. Implicitly in 2. Hence, for completeness, we indicate the changes for a potential N V x1 ,. Just for notational simplicity, it is better to restrict again to the one-dimensional case.

In this way the expressions look nicer, and the transition to three spatial dimensions can be easily done if necessary. It determines in a selfconsistent way both the Wigner function, associated with how the particles distribute in phase space, and the scalar potential, which in turn describe the forces acting on the particles. Equations 2. For plasmas, frequently decaying or periodic boundary conditions are sufficient.

For nano-devices, the choice of boundary conditions is subtler due to the finite size of the system and the nonlocal character of the Wigner function. Indeed, to compute the integral defining the Wigner function, we need to specify f x, v, 0 in the whole space even when dealing with finite size systems. We refer to the specialized literature for more details [10, 20, 30].

Before seeking some of the consequences of the Wigner—Poisson system, let us recapitulate the steps toward its derivation. First, it is a mean field model with the N-body ensemble Wigner function supposed to be factorisable. In particular, the mean field theory is much less numerically demanding, since it requires the discretization of a space with fewer dimensions. However, since correlations are disregarded, the Wigner— Poisson system does not incorporate collisions. Moreover, no spin or relativistic effects are taken into account in our presentation.

Finally, no magnetic fields were introduced yet. Once the Wigner—Poisson system has been derived, it becomes the natural tool in quantum kinetic theory for plasmas, since it is exactly analog to the Vlasov— Poisson system. Hence, the methods applied to the Vlasov—Poisson system can with some optimism be directly translated to quantum plasmas. Other quantum kinetic treatments for assemblies of charged particle systems are obviously important, but cannot compete with the Wigner formalism in the quantum plasma context.

Nevertheless, the simplifications underlying the Wigner—Poisson model points to the relevance of the alternative approaches toward a more sophisticated modeling. However, these developments are outside the scope of the present text. It is instructive to analyze the semiclassical limit of the quantum Vlasov equation 2.

If no quantum effects were present, 2. From the Wigner function, one can compute macroscopic quantities like particle, current and energy densities, very much like in classical physics. Hence, it is a natural trend, to investigate to which extent the methods applied to the Vlasov—Poisson system can be useful in the Wigner—Poisson context. Moreover the positive definiteness of the Wigner function is not preserved by 2. The exception is for linear electric fields, for which the quantum corrections vanishes in 2.

In this case, the Wigner and Vlasov equations coincide to all orders in the nondimensional quantum parameter H. Even in the harmonic oscillator case, when the Wigner and Vlasov equations coincide, f x, v,t cannot be considered as an ordinary probability distribution function. Indeed, not all functions on phase space can be taken as Wigner functions, since a genuine Wigner function should correspond to a positive definite density matrix.

Finally, 2. In addition, the Poisson equation 2. A rigorous proof [29] of the equivalence of 2. However, we can obtain some insight on the interpretation of the ensemble wavefunctions. Quantum statistics effects are not taken into account in the Ansatz 2. However, for simplicity spin considerations will be not included at this moment. Inserting 2. Equation 2. Otherwise, we have a mixed state. Using the map 2. In addition to 2.

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While the direct construction using 2. In particular, when we know we are dealing with a mixed state, what is the minimal number M of ensemble wavefunctions needed? There is no universal answer to these questions in the current literature. In both models, the long-range interactions due to the self-consistent electrostatic potential are assumed to dominate over short-range collisional interactions between two or more particles.

This statement can be made more precise [31]. Correlations between particles or equivalently collisions cannot be neglected if the average potential energy between two electrons become comparable to the average kinetic energy.

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As have been seen in the discussion on the classical and quantum energy coupling parameters of Sect. Hence the Vlasov—Poisson system is the standard model to describe classical electrostatic plasmas in the collisionless approximation. In this case, the N-body Wigner function is expressed as a product of one-particle Wigner functions so that the Wigner—Poisson system is the natural model for collisionless quantum plasmas.

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