A RANDOM CLOUD MODEL FOR THE WIGNER EQUATION

. A probabilistic model for the Wigner equation is studied. The model is based on a particle system with the time evolution of a piecewise deterministic Markov process. Each particle is characterized by a real-valued weight, a position and a wave-vector. The particle position changes continuously, according to the velocity determined by the wave-vector. New particles are created randomly and added to the system. The main result is that appropriate functionals of the process satisfy a weak form of the Wigner equation.

1. Introduction. The following facts were established by Eugene Wigner in 1932 [18]. Let ψ be a solution of the Schrödinger equation where i is the imaginary unit, is Planck's constant (divided by 2π), m is mass, ∆ denotes the Laplace operator, V is potential energy, and R d is the d-dimensional Euclidean space. Consider the function where the symbols * and · denote the complex conjugate and the scalar product, respectively. Under some restrictions on ψ , the function f has the property Moreover, it satisfies the equation where and ∇ denotes the gradient. The function f is real-valued, but not necessarily non-negative.
Existence and uniqueness issues for the Wigner equation (1) were studied, e.g., in [7,6]. In recent years there has been a growing interest in modelling quantum transport in nanoelectronic devices [11]. In this context, the Wigner equation turned out to be convenient, since it can be coupled easily to the scattering part of the semiconductor Boltzmann equation. A Wigner Monte Carlo method has been developed (see, e.g., [8,15]), which is based on random systems of positive and negative particles. Several algorithms used in that method are covered by the model introduced in this paper. More specific comments will be given in the concluding section.
This paper is concerned with the construction of a probabilistic model for the Wigner equation (1). The model is based on a particle system with the time evolution of a piecewise deterministic Markov process [2]. Each particle is characterized by a real-valued weight, a position x ∈ R d and a wave-vector k ∈ R d . The particle position changes continuously, according to the velocity determined by the wave-vector. New particles are created randomly and added to the system. The main result is that appropriate functionals of the process satisfy a weak form of the Wigner equation. Related ideas have been applied previously to the Schrödinger equation in [17].
The paper is organized as follows. The model, the main theorem and the proof are presented in Section 2. Special cases and examples are considered in Section 3. Finally, some comments are given in Section 4.

2.
Results. In this section, first the probabilistic model is introduced. It is based on a particle system with a random time evolution depending on several parameters. Then the main theorem is formulated. It establishes an equation satisfied by certain functionals of the stochastic process. This equation is determined by the parameters of the process. It has a more general form compared to the Wigner equation (1). This generality indicates the properties of the kernel (2), which are essential for the result. Finally, the proof is given. It is based on the theory of piecewise deterministic Markov processes as presented in [2].

Model. Consider a stochastic process
with the state space where the first component represents a weight, the second component is a position vector, and the third component is a wave-vector. Let The time evolution of the process (3) is determined by a flowF and a jump kernel Q . Starting at statez ∈ Z , the process performs a deterministic motion according toF . The random waiting time τ until the next jump satisfies where P denotes the probability measure and Then the process jumps into a new stateκ ∈ Z distributed according to Specification of the flow. We consider the flow so that particles move independently of each other. The single particle flow F is where v(k) denotes the velocity corresponding to k . The flow (9) satisfies the equation Specification of the jump kernel. We consider jump kernels where J(z;z ) = (z 1 , . . . , z N , z 1 , . . . , z k ) ,z = (z 1 , . . . , z k ) ∈ Z , and the kernel q will be specified later. The intensity (6) takes the form A jump according to Q(z, dκ)/λ(z) is generated as follows. With probabilities the j-th particles creates "offspring"z ∈ Z distributed according to The new state isκ where the jump transformation (12) adds the offspring to the system.

2.2.
Theorem. Let the parameters of the stochastic process (3) satisfy the following assumptions.
• The initial stateZ(0) is such that where E denotes mathematical expectation. • The velocity field v(k) is locally Lipschitz continuous. (15) • The offspring-creation kernel q has the following properties.
-The individual offspring-creation rates are bounded, i.e., for some constant C q . -The number of offspring (created during a single jump) is bounded, i.e., for some K = 1, 2, . . . , where The absolute weights of the offspring do not exceed the absolute weight of the parent particle, i.e., where (cf. (5)) Consider some measurable function G on R d × R d × R d and the class M(G) of bounded measurable functions ϕ on R d × R d such that and sup Theorem 2.1. Consider the empirical measures of the process (3), and denote Let the assumptions (14)-(18) be fulfilled. If q is such that for all z = (u, x, k) ∈ Z and ϕ ∈ M(G) (cf. (19)-(21)), then the measure-valued function satisfies the equation for any t ≥ 0 and ϕ ∈ M(G) .
In the following, we provide some straightforward choices for the initial stateZ(0) and the offspring-creation kernel q , which satisfy the assumptions of Theorem 2.1. The special case of the Wigner equation (1), as well as further examples of kernels q , will be discussed in Section 3.
Initial state. Consider a function f 0 ∈ L 1 (R d × R d ) and a probability density π on Let N (0) = 1 . Generate x 1 (0) and k 1 (0) according to π . Under the conditions x 1 (0) = x and k 1 (0) = k , define Then assumption (14) concerning the initial state is satisfied, since The simplest choice is π ∼ |f 0 | . In this case, one obtains Offspring-creation kernel. The main assumption (24) concerning the offspring-creation kernel q is satisfied for the choice Assumptions (17) (with K = 1) and (18) are also fulfilled. Assumption (16) takes the form and provides a restriction on G . According to the kernel (28), a particle z = (u, x, k) creates one offspring wherek is distributed according to the density 1 q(z, Z) |G(x, k,k)| .
Conservation property. If the function G satisfies (29) and then the test function ϕ = 1 belongs to M(G) and equation (26) implies Step 1. Assumptions (16) and (17) imply that, for anyz ∈ Z , the "standard" conditions are satisfied: • the intensity (13) is measurable and such that • the process is regular, i.e., Ez #{l : where (T l ) is the sequence of jump times, #B denotes the number of elements in a set B , and Ez is the conditional expectation with respect to the initial state. Indeed, it follows from (16) that which implies (32). According to (17), the expected number of jumps of the process (3) does not exceed the expected number of jumps of the pure jump process with jumps N → N + K and intensityλ(N ) = C q N . Regularity for the pure jump process follows from the criterion (cf., e.g., [1, p.337]) where ζ l = N + l K is the embedded Markov chain. It follows from (33) and the estimate Step 2. According to [2,Theorem 26.14], the domain D(A) of the extended generator of the process (3) consists of all measurable functions Ψ such that, for anȳ z ∈ Z , t → Ψ(F (t,z)) is absolutely continuous and for some sequence of stopping times σ n ∞ . The extended generator has the form (cf. (10)) For any Ψ ∈ D(A) , the process is a local martingale. If Ez sup and Ez sup then the process (39) is a martingale and one obtains the Dynkin formula then (42) implies Step 3. Consider functions of the form where ψ is measurable. We check Ψ ∈ D(A) provided that • ψ(u, x, k) is differentiable with respect to x and • ψ is either bounded or has the form where ϕ is bounded. Condition (36) is satisfied, according to (8) and (9). In order to check condition (37), we note that (cf. (7), (11)) wherez (l) denotes the offspring created at jump time T l , and (cf. (12)) If ψ is bounded, then condition (37) (with σ n = n) is a consequence of (33), (48) and assumption (17). If ψ has the form (46), then one obtains where . ∞ denotes the sup-norm and According to (49), one obtains Note that each particle created at some jump time T l ≤ t belongs to the system (3) at time t , and the norm of its weight remains constant. According to (9) and assumption (18), one obtains Condition (37) (with σ n = n) is a consequence of (35), (48), (51) and (52).
3. Examples. In this section, first the special case of the Wigner equation (1) is discussed. Then two classes of offspring-creation kernels are introduced, which satisfy the assumptions of Theorem 2.1. One class is based on the explicit form of the function V W , while the other class uses the function V directly. Finally, some specific models are considered, where only a single particle with variable weight is involved.

Wigner equation. For the choices
and (cf. (2)) equation (26) is a measure-valued (and integrated with respect to time) version of the equation which is a weak form of the Wigner equation (1). Assumption (15) is satisfied. Condition (29) takes the form I do not know if there are any non-trivial choices of V , for which condition (60) is satisfied. At least, it does not hold in general, as Example 3.1 illustrates.
Therefore, we introduce the functions where c > 0 is a cutoff parameter. It follows from the definition (2) that V W is anti-symmetric with respect to k , i.e.
and such that sup Thus, the functions (61) are anti-symmetric with respect to k and satisfy provided that V is integrable. Since condition (29) is fulfilled for the function

3.2.
Offspring-creation according to V W . Consider where the function g satisfies and This covers the choice (64) (cf. (62), (63)), which corresponds to the Wigner equation. It follows from (66) that and where g + and g − denote the positive and negative parts of the function g .

3.3.
Offspring-creation according to V . Since (cf. (2)) the function (64) takes the form (cf. (61)) where χ B denotes the indicator function of a set B , and (77) Creation of one particle. The kernel satisfies the main assumption (24) of Theorem 2.1. Assumptions (17) (with K = 1) and (18) are also fulfilled. Assumption (16) takes the form and is satisfied, if V is integrable. Note that According to the kernel (78), a particle z = (u, x, k) creates one offspring, which can be generated as follows (cf. (81)).
Example 3.6. The offspring is wherek andx are distributed according to Creation of two particles. The kernel where u is defined in (79), satisfies the main assumption (24) of Theorem 2.1 (cf. (81)). Assumptions (17) (with K = 2) and (18) are also fulfilled. Assumption (16) is a consequence of (80), sincẽ According to the kernel (83), a particle z = (u, x, k) creates two offspring, which can be generated as follows.
3.4. Single particle models. Consider G of the form (65). The kernel 2 δ ((−u,x,k),(u sign g(x,k),x,k+k),(u sign g(x,k),x,k+k),(u sign g(x,k),x,k+k)) (dz) + δ ((−u,x,k),(u,x,k),(u,x,k),(u,x,k)) (dz) satisfies the main assumption (24) of Theorem 2.1. Indeed, one obtains Assumptions (17) (with K = 4) and (18) are also fulfilled. Assumption (16) is a consequence of (67). Thus, this random cloud model with creation of four particles is covered by Theorem 2.1. However, the special structure of the kernel (84) suggests a simple one-particle process. Instead of creating (−u, x, k) and three identical other particles, the parent particle (u, x, k) is replaced by one other particle with triple weight. This oneparticle process is different from the random cloud model, but both processes are equivalent in terms of empirical measures.
More generally, we consider a piecewise deterministic Markov process with the state space (4), the flow (9) and a jump kernelQ . The extended generator (38) takes the form For the process (85), representation (27) takes the form The jump kernel corresponding to the offspring-creation kernel (84) is Note that (cf. (66), (68)) and (cf. (69))Q 1 (z, Z) = γ(x) = q 4 (z, Z) . The jump can be generated in various equivalent ways.
The kernel (86) can be generalized. Assume that a jump kernel satisfies for some K = 1, 2, . . . . Then the corresponding piecewise deterministic Markov process can be treated via Theorem 2.1. Instead of replacing a particle z = (u, x, k) by a new particle z = (u , x , k ) , a set of offspring is added to the system. This set consists of K identical particles (u /K, x , k ) and the "anti-particle" (−u, x, k) . However, a direct proof (without making use of Theorem 2.1) would also be rather straightforward. Assumption (24) takes the form For the kernel where c 1 , c 2 , c 3 , α, β > 0 , the left-hand side of (88) is

Thus, condition (88) is satisfied if
Note that (cf. (69))Q 4. Comments. Theorem 2.1 presents a probabilistic model for the Wigner equation. The approach is based on the general idea of weighted particle systems.
• In classical direct simulation methods, a non-negative function f , which satisfies a certain equation, is approximated via where y j (t) , j = 1, . . . , N (t) , is a particle system and ϕ is an appropriate test function. The approximation parameter N is related, for example, to the initial number of particles, or to the average number influx per unit time. The variable y represents the state of a single particle. It consists, for example, of position and velocity in the case of the Boltzmann equation, and of position and size in the case of Smoluchowski's coagulation equation.
• If the function f may take negative or even complex values, then a straightforward generalization of (92) is where the weights u j are real-or complex-valued and change in time. Various approaches based on approximations of the form (93) have been studied in the literature. For example, positive variable weights were used for the Boltzmann equation in [13,14] and for Smoluchowski's coagulation equation in [10,9]. Positive and negative weights were used in the deviational particle approach for the Boltzmann equation, e.g., in [5,16,12]. Complex-valued weights were used in the random cloud approach for the Schrödinger equation in [17]. The basic idea of the random cloud approach can be illustrated in an abstract setting without referring to the Wigner equation. Consider a pure jump process with the infinitesimal generator and J(z, z ) = (z 1 , . . . , z N , z ) ,z = (z 1 , . . . , z N ) ∈ Z = ∪ ∞ j=1 Z j , z ∈ Z . According to (94), particles create offspring that are added to the system. The offspring distribution is determined by q . Dynkin's formula, and q(u, y, du , dy ) = |K(y, y )| δ u signK(y,y ) (du ) dy , where K is some integrable kernel. According to (96), a particle (u, y) creates an offspring (u , y ) , where the state y is generated according to |K| and the weight u is determined by the sign of K . Since and, after a transformation into the strong form, Equation (1) corresponds to the kernel Considering the right-hand side of (97) as a "creation term" removes any restriction on the sign of the kernel. If K is non-negative, then the usual interpretation as a "gain term" is recovered. The corresponding "loss term" is responsible for the simultaneous annihilation of the parent particle. This is achieved via the creation of an offspring with the same state and an opposite sign.
Theorem 2.1 provides the basis for an application of the random cloud model in numerics for the Wigner equation. Indeed, if the initial state is chosen as a system of N independent particles with weights normalized by N , then, by the law of large numbers, corresponding functionals of the empirical measures converge (as N → ∞) to the expectation at the right-hand side of the representation (27).
• Some special cases covered by the random cloud model appeared in the literature before. In particular, Examples 3.3, 3.5 and 3.9 were derived (in a different way) and used in the context of the so-called Wigner Monte Carlo method (see, e.g., [8,11,15]). We did not discuss specific numerical issues such as the cancellation of particle pairs with opposite signs, the introduction of fictitious jumps, or the splitting of transport and jump mechanisms. We only mention that in the models with creation of two particles the conservation property (cf. (31)) holds almost surely and not only on average. This makes the corresponding numerical methods much more stable. • Other particular cases, which seem to be new, may also be of interest in numerical applications. Example 3.4 uses the function |V W | instead of V + W , which might be convenient. Examples 3.6 and 3.7 are based on the function V . This might be useful, when the function V W is not explicitly known. The corresponding processes are also interesting from a conceptual point of view, since they show more directly the influence of V on the random behaviour of the particle system. Theorem 2.1 is formulated for the measure-valued version (26) of the Wigner equation. The paper does not address analytical issues such as uniqueness of solutions, existence of densities and regularity properties. These topics are not essential for our main goal -to provide a probabilistic model for the Wigner equation. The random cloud approach seems to be applicable to various extensions of the Wigner equation as, for example, the Wigner-Fokker-Planck equation (cf. [3] and references therein). This case does not fit the framework of piecewise deterministic Markov processes. However, under appropriate assumptions on the jump intensity, it should be possible to proceed in analogy with the treatment of the inelastic Boltzmann equation in [4]. Another interesting open issue is the study of the limiting behaviour for c → ∞ , where c is the cutoff parameter (cf. (61)). It is plausible that lim c→∞ ϕ(x, k) f (c) (t, dx, dk) = ϕ(x, k) f (t, dx, dk) , where the solutions of the Wigner equation with and without cutoff are denoted by f (c) and f , respectively. However, exchanging the limit lim c→∞ and the expectation E at the right-hand side of (27) does not seem to be possible, since the intensity of jumps in the corresponding piecewise-deterministic Markov process would become not only unbounded, but infinite.