Stochastic variational principles for the collisional Vlasov-Maxwell and Vlasov-Poisson equations
aa r X i v : . [ m a t h - ph ] F e b Stochastic variational principles for the collisional Vlasov-Maxwelland Vlasov-Poisson equations
Tomasz M. Tyranowski ∗1,21
Max-Planck-Institut für PlasmaphysikBoltzmannstraße 2, 85748 Garching, Germany Technische Universität München, Zentrum MathematikBoltzmannstraße 3, 85748 Garching, Germany
Abstract
In this work we recast the collisional Vlasov-Maxwell and Vlasov-Poisson equations as sys-tems of coupled stochastic and partial differential equations, and we derive stochastic variationalprinciples which underlie such reformulations. We also propose a stochastic particle method forthe collisional Vlasov-Maxwell equations and provide a variational characterization of it, whichcan be used as a basis for a further development of stochastic structure-preserving particle-in-cellintegrators.
The collisional Vlasov equation ∂f∂t + v ⋅ ∇ x f + qm ( E + v × B ) ⋅ ∇ v f = C [ f ] (1.1)describes the time evolution of the particle density function f = f ( x , v , t ) of plasma consisting ofcharged particles of charge q and mass m which undergo collisions described by the collision operator C [ f ] , and are subject to the electric E = E ( x , t ) and magnetic B = B ( x , t ) fields. The vectors x = ( x , x , x ) and v = ( v , v , v ) denote positions and velocities, respectively. For simplicity,we restrict ourselves to one-spiece plasmas. Usually, the particle density function is normalized,so that the total number of particles is N tot = ∬ f ( x , v , t ) d v d x . However, in this work wewould like to treat f as a probability density function, and therefore we will use the normalization ∬ f ( x , v , t ) d v d x = instead. A self-consistent model of plasma is obtained by coupling (1.1) withthe Maxwell equations ∗ [email protected] x ⋅ E = ρ, (1.2a) ∇ x ⋅ B = , (1.2b) ∇ x × E = − ∂ B ∂t , (1.2c) ∇ x × B = ∂ E ∂t + J , (1.2d)where ρ ( x , t ) = qN tot ∫ R f ( x , v , t ) d v , J ( x , t ) = qN tot ∫ R v f ( x , v , t ) d v , (1.3)denote the charge density and the electric current density, respectively, and the factor N tot is due toour normalization. The system (1.1)-(1.3) is usually referred to as the Vlasov-Maxwell equations.It will also be convenient to express the electric and magnetic fields in terms of the scalar ϕ ( x , t ) and vector A ( x , t ) potentials E = −∇ x ϕ − ∂ A ∂t , (1.4a) B = ∇ x × A , (1.4b)as is typical in electrodynamics. The Vlasov-Poisson equations are an approximation of the Vlasov-Maxwell equations in the nonrelativistic zero-magnetic field limit (see Section 6). The main goalof this work is to provide a variational characterization of the Vlasov-Maxwell and Vlasov-Poissonequations via a stochastic Lagrange-d’Alembert type of a principle.Variational principles have proved extremely useful in the study of nonlinear evolution partialdifferential equations (PDEs). For instance, they often provide physical insights into the problembeing considered; facilitate discovery of conserved quantities by relating them to symmetries viaNoether’s theorem; allow one to determine approximate solutions to PDEs by minimizing the actionfunctional over a class of test functions (see, e.g., [15]); and provide a way to construct a class ofnumerical methods called variational integrators (see [60], [61]). A variational principle for thecollisionless Vlasov-Maxwell equations was first proposed in [58]. It has been used to derive variousparticle discretizations of the Vlasov-Maxwell and Vlasov-Poisson equations (see [24], [56], [57], [72],[76]), including structure-preserving variational particle-in-cell methods ([75], [88], [89]). It has alsobeen applied to gyrokinetic theory (see, e.g., [7], [79]). For other formulations and extensions seealso [90].A structure-preserving description of collisional effects is far less developed. A metriplecticframework for the Vlasov-Maxwell-Landau equations has been presented in [32] and [46]. Morerecently, a stochastic variational principle has been proposed in [48] to describe collisional effectsfor the Vlasov equation with a fixed external electric field. To the best of our knowledge, to dateno variational principle has been derived for the collisional Vlasov-Maxwell and Vlasov-Poissonequations. In this work we extend the notion of the stochastic Lagrange-d’Alembert principlepresented in [48] to plasmas evolving in self-consistent electromagnetic fields. The main idea of ourapproach is to interpret the Vlasov equation (1.1) as a Fokker-Planck equation and consider theassociated stochastic differential equations. 2he idea of using stochastic differential equations to model collisions has been pursued by anumber of authors over the last few decades; see, e.g., [1], [6], [11], [14], [21], [25], [27], [31], [43],[48], [51], [59], [63], [73], [74], [92].There has been an ever growing body of literature dedicated to stochastic variational principlesin recent years. Stochastic variational principles allow the introduction of noise into systems in sucha way that the resulting probabilistic models retain all or some of the geometric properties of theirdeterministic counterparts. For this reason stochastic variational principles have been considered inthe context of Lagrangian and Hamiltonian mechanics ([5], [8], [9], [10], [38], [48], [50], [83]), solitondynamics ([36], [37]), fluid dynamics ([2], [13], [16], [17], [18], [29], [34], [35]), and kinetic plasmatheory ([48]). Main content
The main content of the remainder of this paper is, as follows.In Section 2 we recast the collisional Vlasov-Maxwell equations as a system of coupled stochasticand partial differential equations.In Section 3 we discuss the relationship between particle methods and stochastic modeling. Weformulate a stochastic particle discretization for the collisional Vlasov-Maxwell equations andcast it in a form that allows the derivation of a variational principle.In Section 4 we describe the variational structure underlying the stochastic particle discretizationof the Vlasov-Maxwell system. The main result of this section is Theorem 4.2, in which astochastic Lagrange-d’Alembert principle for the particle discretization is proved.In Section 5 we generalize the ideas from Section 4 to the original undiscretized equations. Themain result of this section is Theorem 5.1, in which a stochastic Lagrange-d’Alembert principleis proved for a special case of the collisional Vlasov-Maxwell equations.In Section 6 we prove a stochastic Lagrange-d’Alembert principle applicable to the Vlasov-Poissonequations. The main result of this section is Theorem 6.1.Section 7 contains the summary of our work.
Various collision models and various forms of the collision operator C [ f ] are considered in theplasma physics literature (see, e.g., [12], [62]). A key step towards a stochastic variational principleis a probabilistic interpretation of the Vlasov equation (1.1). Therefore, in this work we will beinterested only in those collision operators for which (1.1) takes the form of a linear or stronglynonlinear Fokker-Planck equation (see, e.g., [26], [28], [69]). Namely, we will assume that thecollision operator can be expressed as C [ f ] = ∑ i,j = ∂ ∂v i ∂v j [ D ij ( x , v ; f ) f ] − ∑ i = ∂∂v i [ K i ( x , v ; f ) f ] , (2.1)3or some symmetric positive semi-definite matrix D ij ( x , v ; f ) and vector K i ( x , v ; f ) functions, wherethe dependence of D ij and K i on f may in general be nonlinear, and may involve differential andintegral forms of f . In that case (1.1) is an integro-differential equation, the so-called stronglynonlinear Fokker-Planck equation (see [26]). In case D ij and K i are independent of f , that is, D ij ( x , v ; f ) = D ij ( x , v ) and K i ( x , v ; f ) = K i ( x , v ) , the Vlasov equation (1.1) reduces to the stan-dard linear Fokker-Planck equation. We will further assume that D ij and K i can be expressed inthe form D ij ( x , v ; f ) = M ∑ ν = g iν g jν , K i ( x , v ; f ) = G i + M ∑ ν = ∑ j = ∂g iν ∂v j g jν , (2.2)for a vector function G ( x , v ; f ) , and a family of vector functions g ν ( x , v ; f ) with ν = , . . . , M .Note that given a symmetric positive semi-definite matrix D ij , a decomposition (2.2) can alwaysbe found, but it may not be unique. For instance, one may take M = and assume that g iν = g νi for i, ν = , , . Then the first equation in (2.2) implies that the family of functions g iν can bedetermined by calculating the square root of the matrix D ij , and the second equation in (2.2) canbe used to calculate the function G . If (1.1) has the form of a Fokker-Planck equation, then theparticle density function f can be interpreted as the probability density function for a stochasticprocess ( X ( t ) , V ( t )) ∈ R × R . This stochastic process then satisfies the Stratonovich stochasticdifferential equation (see [26], [28], [44], [69]) d X = V dt, (2.3a) d V = ( qm E ( X , t ) + qm V × B ( X , t ) + G ( X , V ; f )) dt + M ∑ ν = g ν ( X , V ; f ) ○ dW ν ( t ) , (2.3b)where W ( t ) , . . . , W M ( t ) denote the components of the standard M -dimensional Wiener process,and ○ denotes Stratonovich integration. Note that the terms G and g ν can be interpreted asexternal forces, and that in their absence the equations (2.3) reduce to the equations of motionof a charged particle in an electromagnetic field. We will therefore refer to G and g ν as forcingterms. The electric and magnetic fields are coupled via the Maxwell equations (1.2). It shouldalso be noted that unless (1.1) is linear, the right-hand side of (2.3) depends on f . In order toobtain a self-consistent system, one can express f in terms of the stochastic processes X and V as f ( x , v , t ) = E [ δ ( x − X ( t )) δ ( v − V ( t ))] , where E denotes the expected value, and δ is Dirac’s delta.This can be further plugged into (1.3). Together, we get f ( x , v , t ) = E [ δ ( x − X ( t )) δ ( v − V ( t ))] , (2.4a) ρ ( x , t ) = qN tot E [ δ ( x − X ( t ))] , (2.4b) J ( x , t ) = qN tot E [ V ( t ) δ ( x − X ( t ))] . (2.4c)Equations (1.2), (2.3), and (2.4) form a self-consistent system of stochastic and partial differentialequations whose solutions are the stochastic processes X ( t ) , V ( t ) , and the functions E ( x , t ) , B ( x , t ) .4 emark. Upon substituting (2.4a), the forcing terms G and g ν become functionals of the pro-cesses X and V , that is, G ( x , v ; f ) = G ( x , v ; X , V ) and g ν ( x , v ; f ) = g ν ( x , v ; X , V ) . However,for convenience and simplicity, throughout this work we will stick to the notation G ( x , v ; f ) and g ν ( x , v ; f ) , understanding that the probability density is given by (2.4a) (or by (3.2a) for particlediscretizations; see Section 3). Below we list a few examples of collision operators that fit the decription presented in Section 2.1.
The Lenard-Bernstein collision operator, C [ f ] = ν c ( µ ∇ v ⋅ ( v f ) + γ v f ) , (2.5)where ν c > , µ > , and γ > are parameters, models small-angle collisions, and was originallyused to study longitudinal plasma oscillations (see [12], [52], [62]). It can be easily verified that anexample decomposition (2.2) for M = is given by the functions G ( x , v ) = − ν c µ v , g ( x , v ) = ⎛⎜⎝√ ν c γ ⎞⎟⎠ , g ( x , v ) = ⎛⎜⎝ √ ν c γ ⎞⎟⎠ , g ( x , v ) = ⎛⎜⎝ √ ν c γ ⎞⎟⎠ . (2.6)Note that these functions do not explicitly depend on f , therefore in this case (1.1) is a linearFokker-Planck equation. The Lorentz collision operator models electron-ion interactions via pitch-angle scattering and isgiven by the formula C [ f ] = ν c (∣ v ∣) ∇ v ⋅ (∣ v ∣ I − v ⊗ v ) ∇ v f, (2.7)where ν c (∣ v ∣) is the collisional frequency as a function of the absolute value of velocity, I is the × identity matrix, and ⊗ denotes tensor product. The primary effect of this type of scatteringis a change of the direction of the electron’s velocity with negligible energy loss. More informationabout the Lorentz collision operator, including the exact form of the collision frequency, can befound in, e.g., [3], [12], [42], [62]. It can be verified by a straightforward calculation that an exampledecomposition (2.2) for M = is given by the functions G ( x , v ) = , g ( x , v ) = √ ν c (∣ v ∣) ⎛⎜⎝ − v v ⎞⎟⎠ , g ( x , v ) = √ ν c (∣ v ∣) ⎛⎜⎝ v − v ⎞⎟⎠ , g ( x , v ) = √ ν c (∣ v ∣) ⎛⎜⎝ − v v ⎞⎟⎠ . (2.8)5ote that these functions do not explicitly depend on f , therefore also in this case (1.1) is a linearFokker-Planck equation. The more general Coulomb collision operator has the form (2.1) with D ij ( x , v ; f ) = N tot Γ ∫ R ∣ v − u ∣ δ ij − ( v i − u i )( v j − u j )∣ v − u ∣ f ( x , u , t ) d u ,K i ( x , v ; f ) = − N tot Γ ∫ R v i − u i ∣ v − u ∣ f ( x , u , t ) d u , (2.9)where N tot appears due to our normalization of f , δ ij is Kronecker’s delta, and Γ = ( πq / m ) ln Λ ,with ln Λ denoting the so-called Coulomb logarithm. The Coulomb operator describes collisions inwhich the fundamental two-body force obeys an inverse square law, and makes the assumption thatsmall-angle collisions are more important that collisions resulting in large momentum changes (see[12], [62], [70]). A decomposition (2.2) can be found, for example, via the procedure outlined inSection 2.1. However, the expressions for G and g ν are complicated, therefore we are not statingthem here explicitly. Note that D ij and K i explicitly depend on f . Therefore, for the Coulomboperator the Vlasov equation (1.1) is a strongly nonlinear Fokker-Planck equation. Note also that D ij and K i can be explicitly written as functionals of the stochastic processes X and V as D ij ( x , v ; X , V ) = N tot Γ ⋅ E [ ∣ v − V ( t )∣ δ ij − ( v i − V i ( t ))( v j − V j ( t ))∣ v − V ( t )∣ δ ( x − X ( t ))] ,K i ( x , v ; X , V ) = − N tot Γ ⋅ E [ v i − V i ( t )∣ v − V ( t )∣ δ ( x − X ( t ))] . (2.10) Particle modelling is one of the most popular numerical techniques for solving the Vlasov equa-tion (see, e.g., [4], [33]). In this section we discuss the connections between particle methods andstochastic modelling.The standard particle method for the collisionless Vlasov equation (1.1) (with C [ f ] = ) consistsof substituting the Ansatz f ( x , v , t ) = ∑ Na = w a δ ( x − X a ( t )) δ ( v − V a ( t )) for the particle densityfunction, and deriving the corresponding ordinary differential equations satisfied by the ‘particle’positions X a ( t ) and velocities V a ( t ) , which turn out to be the characteristic equations. Note thatwe did a qualitatively similar thing in Section 2.1, where we turned the original collisional Vlasovequation into the system of stochastic differential equations (2.3), which in the absence of the forcingterms G and g ν have the same form as the characteristic equations, and in fact the ‘particles’ X a ( t ) and V a ( t ) can be interpreted as realizations of the stochastic processes X ( t ) and V ( t ) for differentelementary events ω ∈ Ω .When the right-hand side of (2.3) does not depend on f , then (2.3) can in principle be solvednumerically with the help of any standard stochastic numerical method (see, e.g., [44]), and each6ealization of the stochastic processes can be simulated independently of others. When the right-hand side of (2.3) depends on f , then all realizations of the stochastic processes have to be solvedfor simultaneously, so that at each time step the probability density function f can be numericallyapproximated (see, e.g., [26]). Such an approach, however, does not quite lend itself to a geometricformulation. Therefore, in order to be able to introduce a variational principle in Section 4, letus consider N stochastic processes X , V , . . . , X N , V N , with each pair ( X a , V a ) satisfying thestochastic differential system d X a = V a dt, (3.1a) d V a = ( qm E ( X a , t ) + qm V a × B ( X a , t ) + G ( X a , V a ; f )) dt + M ∑ ν = g ν ( X a , V a ; f ) ○ dW νa ( t ) , (3.1b)for a = , . . . , N , where W a = ( W a , . . . , W Ma ) are N independent M -dimensional Wiener processes.Note that the systems (3.1) are decoupled from each other for different values of a , and each systemis driven by an independent Wiener process W a . Therefore, the pairs ( X a , V a ) for a = , . . . , N areindependent identically distributed (i.i.d.) stochastic processes, each with the probability densityfunction f that satisfies the original Fokker-Planck equation (1.1). In that sense (3.1) is equiv-alent to (2.3). The advantage is that instead of considering N realizations of the 6-dimensionalstochastic process ( X , V ) in (2.3), one can consider one realization of the N -dimensional process ( X , V , . . . , X N , V N ) in (3.1). Such a reformulation will allow us to identify an underlying stochas-tic variational principle in Section 4. The last step leading to the stochastic particle discretizationis approximating the probability density function f in (3.1). This can be done with the help of thelaw of large numbers, namely, one can approximate (2.4) for large N as f ( x , v , t ) ≈ N N ∑ a = δ ( x − X a ( t )) δ ( v − V a ( t )) , (3.2a) ρ ( x , t ) ≈ qN tot N N ∑ a = δ ( x − X a ( t )) , (3.2b) J ( x , t ) ≈ qN tot N N ∑ a = V a ( t ) δ ( x − X a ( t )) . (3.2c)It is easy to see that (3.2a) coincides with the standard Ansatz used in particle modelling (withthe weights w a = / N ). Therefore, the system of stochastic differential equations (3.1) with theapproximation (3.2), and with the electromagnetic field coupled via the Maxwell equations (1.2),can be considered as a stochastic particle discretization of the collisional Vlasov-Maxwell equations. Remark.
Upon substituting (3.2a), the forcing terms G and g ν become functionals of the pro-cesses X , . . . , X N and V , . . . , V N . Similar to the discussion in Section 2.1, for convenience andsimplicity, throughout this work we will stick to the notation G ( x , v ; f ) and g ν ( x , v ; f ) , under-standing that the probability density is given by (3.2a) for particle discretizations.7 Variational principle for the particle discretization
In this section we propose an action functional which can be understood as a stochastic version ofthe Low action functional (see [58]), and we prove a variational principle underlying the particlediscretization introduced in Section 3, akin to the stochastic Lagrange-d’Alembert principle firstintroduced in [48].
Before we introduce the action functional, we need to identify suitable function spaces on whichit will be defined. For simplicity, let our spatial domain be the whole three-dimensional space R ,and let us consider the time interval [ , T ] for some T > . Let ( Ω , F , P ) be the probability spacewith the filtration { F t } t ≥ , and let W a = ( W a , . . . , W Ma ) for a = , . . . , N denote N independent M -dimensional Wiener processes on that probability space (such that W νa ( t ) is F t -measurable for all t ≥ ). The stochastic processes X a ( t ) and V a ( t ) satisfy (3.1), so they are in particular F t -adaptedsemimartingales, and have almost surely continuous paths (see [68]). We also notice that there isno diffusion term in (3.1a), therefore we even have that the processes X a ( t ) are almost surely ofclass C . We introduce the notation C k Ω ,T = { X ∈ L ( Ω × [ , T ] , R ) ∣ X is a F t -adapted semimartingale, almost surely of class C k } . (4.1)Note that this set is a vector space (see [68]). The potentials ϕ and A satisfy the Maxwell equations(1.2) and (1.4), therefore we require them to be of class C . However, since our spatial domain isunbounded, we further need to assume that the vector fields E and B are square integrable. Weintroduce the notation X ( R n ) = { A ∈ C ( R × [ , T ] , R n ) ∩ L ∞ ( R × [ , T ] , R n ) ∣ ∀ i, j ∶ ∂A i ∂x j , ∂A i ∂t ∈ L ( R × [ , T ])} , X ( R n ) = C ( R × [ , T ] , R n ) , (4.2)where X ( R n ) is simply the space of compactly supported elements of X ( R n ) . Let us consider the action functional S ∶ Ω × ( C ,T ) N × ( C ,T ) N × ( C ,T ) N × X ( R ) × X ( R ) Ð → R (4.3)defined by the formula S [ X , . . . , X N , V , . . . , V N , P , . . . , P N , ϕ, A ] = N tot N N ∑ a = ⎡⎢⎢⎢⎢⎣ ∫ T ( m ∣ V a ∣ − qϕ ( X a , t ) + q V a ⋅ A ( X a , t ) + P a ⋅ ( ˙X a − V a )) dt ⎤⎥⎥⎥⎥⎦ + ∫ T ∫ R (∣ E ∣ − ∣ B ∣ ) d x dt, (4.4)8here ˙X a denotes the time derivative of X a , and the electric and magnetic fields E and B areexpressed in terms of the partial derivatives of the potentials ϕ and A as in (1.4). Followingthe standard convention in stochastic analysis, we will omit writing elementary events ω ∈ Ω asarguments of stochastic processes unless otherwise needed, i.e., X a ( t ) ≡ X a ( ω, t ) . The actionfunctional (4.4) resembles the Low action functional introduced in [58]. In fact, it can be viewedas a particle discretization of the Low action functional, written in terms of stochastic processes(see [24], [56], [72], [75], [88], [89]). The term P a ⋅ ( ˙X a − V a ) is the so-called Hamilton-Pontryaginkinematic constraint (see, e.g., [49], [91]) that enforces that ˙X a = V a using the Lagrange multiplier P a , which turns out to be the conjugate momentum. In principle, this constraint is not necessaryin our context—we could omit it and replace V a with ˙X a in (4.4). We will, however, keep it inorder to make a clear connection with the theory developed in [9]. It also makes the notation in theproof of the stochastic Lagrange-d’Alembert principle in Section 4.3 more convenient and elegant.Note that the action functional S is itself a random variable, as ω ∈ Ω is one of its arguments.We will define the variation of S with respect to the variation δ X a ∈ C ,T of the argument X a as δ X a S = ddǫ ∣ ǫ = S [ X , . . . , X a + ǫδ X a , . . . , X N , V , . . . , V N , P , . . . , P N , ϕ, A ] . (4.5)Since the potentials ϕ and A are C , and the processes X b , V b , and P b are almost surely continuous,we can use a dominated convergence argument to interchange the differentiation with respect to ǫ and integration with respect to t to obtain δ X a S = N tot N ∫ T ( − q ∇ x ϕ ( X a , t ) ⋅ δ X a + q ∑ i,j = V j ∂A j ∂x i ( X a , t ) δX ia + P a ⋅ δ ˙X a ) dt. (4.6)Since δ X a is almost surely differentiable, we have that its stochastic differential is simply dδ X a = δ ˙X a dt . Furthermore, both δ X a and P a are almost surely continuous semimartingales, thereforeusing the integration by parts formula for semimartingales (see [68]) we can write ∫ T P a ⋅ δ ˙X a dt = ∫ T P a ○ dδ X a = P a ( t ) ⋅ δ X a ( t )∣ T − ∫ T δ X a ○ d P a , (4.7)where the Stratonovich integrals are understood in the sense that ∫ δ X a ○ d P a = ∑ i ∫ δX ia ○ dP ia . Bysubstituting (4.7) in (4.6), we obtain δ X a S = N tot N ( P a ( T ) ⋅ δ X a ( T ) − P a ( ) ⋅ δ X a ( )) + N tot N ⎡⎢⎢⎢⎢⎣ − ∫ T δ X a ○ d P a + ∫ T ( − q ∇ x ϕ ( X a , t ) ⋅ δ X a + q ∑ i,j = V j ∂A j ∂x i ( X a , t ) δX ia ) dt ⎤⎥⎥⎥⎥⎦ . (4.8)Variations with respect to δ V a , δ P a ∈ C ,T are defined analogously to (4.5). Similar computationsyield (note that integration by parts is not necessary)9 V a S = N tot N ∫ T ( m V a + q A ( X a , t ) − P a ) ⋅ δ V a dt, (4.9) δ P a S = N tot N ∫ T ( ˙X a − V a ) ⋅ δ P a dt. (4.10)The variation of S with respect to the variation δ A ∈ X ( R ) of the vector potential A is defined as δ A S = ddǫ ∣ ǫ = S [ X , . . . , X N , V , . . . , V N , P , . . . , P N , ϕ, A + ǫδ A ] . (4.11)Switching the order of differentiation and integration, integrating by parts, and using the fact that δ A is compactly supported, one arrives at δ A S = ∫ T ∫ R ( J + ∂ E ∂t − ∇ x × B ) ⋅ δ A d x dt − ∫ R ( E ( x , T ) ⋅ δ A ( x , T ) − E ( x , ) ⋅ δ A ( x , )) d x , (4.12)where in the derivations we have used (3.2c) and N tot N N ∑ b = [ ∫ T q V b ( t ) ⋅ δ A ( X b , t ) dt ] = ∫ T ∫ R qN tot N N ∑ b = [ q V b ( t ) δ ( x − X b ( t ))] ⋅ δ A ( x , t ) d x dt = ∫ T ∫ R J ( x , t ) ⋅ δ A ( x , t ) d x dt, (4.13)and the remaining calculations are standard, and can be found in, e.g., [23], [39]. The variation of S with respect to the variation δϕ ∈ X ( R ) of the scalar potential ϕ is defined in a similar fashion,and after similar calculations one obtains δ ϕ S = ∫ T ∫ R ( ∇ x ⋅ E − ρ ) ⋅ δϕ d x dt, (4.14)where ρ is given by (3.2b). The total variation of S with respect to the variations of all argumentsis given by δS = N ∑ a = ( δ X a S + δ V a S + δ P a S ) + δ ϕ S + δ A S. (4.15) While the standard rules of the calculus of variations apply to the variations (4.12) and (4.14), thevariations (4.8), (4.9), (4.10) involve stochastic processes and stochastic integrals. Therefore, beforewe can formulate a stochastic variational principle, we need to prove the following lemma.10 emma 4.1.
Let X ∈ C ,T and V , P ∈ C ,T , and let R , r ν ∶ R × R Ð→ R be of class C for ν = , . . . , M . Then ∀ Z ∈ C ,T ∶ ∫ T ( Z ( t ) ○ d P − R ( X , V ) ⋅ Z ( t ) dt − M ∑ ν = r ν ( X , V ) ⋅ Z ( t ) ○ dW ν ( t )) = a.s. (4.16) if and only if for all t ∈ [ , T ] ∀ t ∈ [ , T ] ∶ ∫ t ( d P ( τ ) − R ( X ( τ ) , V ( τ )) dτ − M ∑ ν = r ν ( X ( τ ) , V ( τ )) ○ dW ν ( τ )) = a.s. (4.17) Proof.
Suppose that (4.17) holds. Then (4.16) follows from the associativity property of theStratonovich integral (see, e.g., the proof of Theorem 2.1 in [38]). Conversely, assume that (4.16)is satisfied, and let us prove that (4.17) follows. Our reasoning very closely follows the proof ofTheorem 3.3 in [9]. Pick any time t ∈ [ , T ] . We will use e , e , and e to denote the standardCartesian basis vectors for R . Pick a basis vector e i . The condition (4.16) in particular holds for Z ’s which are C functions of time, i.e., non-random. The main idea of the proof is to construct aone-parameter family of C functions Z ǫ which converge to [ ,t ] e i as ǫ Ð→ , and show that theintegral in (4.16) converges almost surely to the integral in (4.17). Let us introduce the notation I ( X , V , P , Z ) = ∫ T ( Z ( τ ) ○ d P ( τ ) − R ( X , V ) ⋅ Z ( τ ) dτ − M ∑ ν = r ν ( X , V ) ⋅ Z ( τ ) ○ dW ν ( τ )) , (4.18) I ∗ ( X , V , P ) = ∫ T ( [ ,t ] e i ○ d P ( τ ) − R ( X , V ) ⋅ [ ,t ] e i dτ − M ∑ ν = r ν ( X , V ) ⋅ [ ,t ] e i ○ dW ν ( τ )) = ∫ t ( dP i ( τ ) − R i ( X ( τ ) , V ( τ )) dτ − M ∑ ν = r iν ( X ( τ ) , V ( τ )) ○ dW ν ( τ )) . (4.19)Define the functions h ∶ [ , ǫ ] Ð→ [ , ] and h ∶ [ t − ǫ, t ] Ð→ [ , ] by the formulas h ( τ ) = τǫ − τ ǫ , h ( τ ) = ⎧⎪⎪⎨⎪⎪⎩ − ǫ ( τ − t + ǫ ) + if t − ǫ ≤ τ ≤ t − ǫ , ǫ ( τ − t + ǫ ) − ǫ ( τ − t + ǫ ) + if t − ǫ < τ ≤ t . (4.20)Note that h ( ) = h ( t ) = , h ( ǫ ) = h ( t − ǫ ) = , and h ′ ( ǫ ) = h ′ ( t − ǫ ) = h ′ ( t ) = . Define furtherthe family of functions Z ǫ by the formula Z ǫ ( τ ) = ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ h ( τ ) e i if ≤ τ ≤ ǫ , e i if ǫ < τ < t − ǫ , h ( τ ) e i if t − ǫ ≤ τ ≤ t , if t < τ ≤ T . (4.21)11t is easy to see that Z ǫ is continuously differentiable on [ , T ] , and converges to [ ,t ] e i in the L norm as ǫ goes to zero. Using (4.18), (4.19), (4.20), and (4.21), we have I ∗ ( X , V , P ) − I ( X , V , P , Z ǫ ) =∫ ǫ (( − h ( τ )) ○ dP i ( τ ) − ( − h ( τ )) R i ( X , V ) dτ − M ∑ ν = ( − h ( τ )) r iν ( X , V ) ○ dW ν ) + ∫ tt − ǫ (( − h ( τ )) ○ dP i ( τ ) − ( − h ( τ )) R i ( X , V ) dτ − M ∑ ν = ( − h ( τ )) r iν ( X , V ) ○ dW ν ) . (4.22)By definition, the Stratonovich integrals in (4.22) can be expressed in terms of the Itô integrals as ∫ ǫ ( − h ( τ )) r iν ( X , V ) ○ dW ν = ∫ ǫ ( − h ( τ )) r iν ( X , V ) dW ν + [( − h ( τ )) r iν ( X , V ) , W ν ( τ )] ǫ , ∫ tt − ǫ ( − h ( τ )) r iν ( X , V ) ○ dW ν = ∫ tt − ǫ ( − h ( τ )) r iν ( X , V ) dW ν + [( − h ( τ )) r iν ( X , V ) , W ν ( τ )] tt − ǫ , (4.23)for each ν = , . . . , M , where [ ⋅ , ⋅ ] denotes the quadratic covariation process. Since the quadraticcovariation of almost surely continuous semimartingales is itself a semimartingale with almost surelycontinuous paths (see Theorem 23 in Chapter II.6 of [68]), we have that [( − h ( τ )) r iν ( X ( τ ) , V ( τ )) , W ν ( τ )] ǫ Ð→ ( − h ( )) r iν ( X ( ) , V ( )) W ν ( ) = a.s. as ǫ Ð→ , (4.24)since W ν ( ) = almost surely. In a similar fashion we show [( − h ( τ )) r iν ( X ( τ ) , V ( τ )) , W ν ( τ )] tt − ǫ Ð→ a.s. as ǫ Ð→ . (4.25)Using (4.22) and (4.23), we have the estimate Note that our definition (4.21) is slightly different from the corresponding definition in [9], because the testfunctions used in [9] are in fact not differentiable at τ = t . This, however, is of little consequence for the rest of theproof. I ∗ ( X , V , P ) − I ( X , V , P , Z ǫ )∣ ≤ ∣ ∫ ǫ (( − h ( τ )) ○ dP i ( τ ) − ( − h ( τ )) R i ( X , V ) dτ − M ∑ ν = ( − h ( τ )) r iν ( X , V ) dW ν )∣´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Γ + ∣ ∫ tt − ǫ (( − h ( τ )) ○ dP i ( τ ) − ( − h ( τ )) R i ( X , V ) dτ − M ∑ ν = ( − h ( τ )) r iν ( X , V ) dW ν )∣´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Γ + M ∑ ν = ∣[( − h ( τ )) r iν ( X , V ) , W ν ( τ )] ǫ ∣ + M ∑ ν = ∣[( − h ( τ )) r iν ( X , V ) , W ν ( τ )] tt − ǫ ∣ . (4.26)By bounding the integrands and using the Itô isometry theorem, it is shown in [9] that Γ Ð→ and Γ Ð→ in mean-square as ǫ Ð→ , and consequently, by invoking the Borel-Cantelli lemma, thereexists a subsequence ( ǫ n ) such that ǫ n Ð→ as n Ð→ ∞ , for which Γ Ð→ and Γ Ð→ almostsurely. Together with (4.24) and (4.25), this means that I ( X , V , P , Z ǫ n ) Ð→ I ∗ ( X , V , P ) almostsurely. Given the assumption (4.16), we have that I ∗ ( X , V , P ) = almost surely, which completesthe proof. Remark.
Equation (4.17) means that P ( t ) , X ( t ) , and V ( t ) satisfy a stochastic differential equa-tion, which can be written in the differential form as d P ( t ) = R ( X ( t ) , V ( t )) dt + M ∑ ν = r ν ( X ( t ) , V ( t )) ○ dW ν ( t ) . (4.27)We are now in a position to formulate and prove a stochastic variational principle that generalizesthe deterministic Lagrange-d’Alembert principle for forced Lagrangian and Hamiltonian systems,akin to the stochastic variational principle introduced in [48]. Theorem 4.2 ( Stochastic Lagrange-d’Alembert principle for particles).
Let X a ∈ C ,T and V a , P a ∈ C ,T for a = , . . . , N be stochastic processes, and let A ∈ X ( R ) , ϕ ∈ X ( R ) be functions.Assume that G ( ⋅ , ⋅ ; f ) and g ν ( ⋅ , ⋅ ; f ) for ν = , . . . , M are C functions of their arguments, where f isgiven by (3.2a) . Then X a , V a , P a , A , and ϕ satisfy the system of stochastic differential equations ˙X a = V a , (4.28a) P a = m V a + q A ( X a , t ) , (4.28b) dP ia = ( − q ∂ϕ∂x i ( X a , t ) + q ∑ j = V ja ∂A j ∂x i ( X a , t ) + m G i ( X a , V a ; f )) dt + m M ∑ ν = g iν ( X a , V a ; f ) ○ dW νa ( t ) , (4.28c) for i = , , and a = , . . . , N , together with the Maxwell equations (1.2) , (1.4) , (3.2) on the timeinterval [ , T ] , if and only if they satisfy the following variational principle S + mN tot N N ∑ a = [ ∫ T G ( X a , V a ; f ) ⋅ δ X a dt + M ∑ ν = ∫ T g ν ( X a , V a ; f ) ⋅ δ X a ○ dW νa ( t )] = (4.29) for arbitrary variations δ X a ∈ C ,T , δ V a , δ P a ∈ C ,T , δ A ∈ X ( R ) , and δϕ ∈ X ( R ) , with δ X a ( ) = δ X a ( T ) = almost surely, and δ A ( x , ) = δ A ( x , T ) = for all x ∈ R , where the action functional S is given by (4.4) .Proof. Let us first consider the variations with respect to A in (4.29). Given the boundary conditionsfor δ A , from the standard calculus of variations we have that δ A S = (see Equation (4.12)) forall δ A if and only if (1.2d) is satisfied. Similarly, δ ϕ S = (see Equation (4.14)) holds for all δϕ if and only if (1.2a) holds. Further, for variations with respect to V a we have that δ V a S = (seeEquation (4.9)) for all δ V a if and only if (4.28b) is satisfied almost surely, which follows from thestandard theorem of the calculus of variations, since the integral in (4.9) is a standard Lebesgueintegral, and the integrands are almost surely continuous. Similarly, δ P a S = (see Equation (4.10))for all δ P a if and only if (4.28a) is satisfied almost surely. Finally, for variations with respect to X a , Equations (4.8) and (4.29) give ∫ T ⎛⎝ − δ X a ○ d P a + ( − q ∇ x ϕ ( X a , t ) ⋅ δ X a + q ∑ i,j = V j ∂A j ∂x i ( X a , t ) δX ia + m G ( X a , V a ; f ) ⋅ δ X a ) dt + m M ∑ ν = g ν ( X a , V a ; f ) ⋅ δ X a ○ dW νa ( t )⎞⎠ = , (4.30)which, by Lemma 4.1, holds for all δ X a if and only if (4.28c) is satisfied. Remark.
Equation (4.28) is expressed in terms of the Lagrange multipliers P a , which, as can beseen in (4.28b), turn out to be the conjugate momenta. The conjugate momenta can be eliminated,and Equation (4.28) can be recast as Equation (3.1b), which is shown in the following theorem. Theorem 4.3.
Equations (3.1) and (4.28) are equivalent.Proof.
By calculating the stochastic differential on both sides of (4.28b) and substituting (4.28a),we obtain dP ia = m dV ia + q ∑ j = V ja ∂A i ∂x j ( X a , t ) dt + q ∂A i ∂t ( X a , t ) dt (4.31)for each i = , , and a = , . . . , N . Comparing this with (4.28c), and using (1.4), one eliminatesthe conjugate momenta and obtains Equation (3.1b).14 emark. Theorems 4.2 and 4.3 provide a variational formulation of the stochastic particle methodfrom Section 3. One can further perform a variational discretization of the electromagnetic fields A and ϕ , for instance along the lines of [75], [77] or [47], thus obtaining a stochastic particle-in-cell(PIC) discretization of the collisional Vlasov-Maxwell equations. The resulting structure-preservingnumerical methods will be investigated in a follow-up work. The form of the action functional (4.4) and of the Lagrange-d’Alembert principle (4.29) suggeststhat it should be possible to formulate a similar variational principle for the stochastic reformulationof the Vlasov-Maxwell system discussed in Section 2.1. In this section we provide a partial answerto this problem.
Let us consider the action functional ¯ S ∶ C ,T × C ,T × C ,T × X ( R ) × X ( R ) Ð→ R (5.1)defined by the formula ¯ S [ X , V , P , ϕ, A ] = N tot ⋅ E ⎡⎢⎢⎢⎢⎣ ∫ T ( m ∣ V ∣ − qϕ ( X , t ) + q V ⋅ A ( X , t ) + P ⋅ ( ˙X − V )) dt ⎤⎥⎥⎥⎥⎦ + ∫ T ∫ R (∣ E ∣ − ∣ B ∣ ) d x dt, (5.2)where ˙X denotes the time derivative of X , the electric and magnetic fields E and B are expressedin terms of the partial derivatives of the potentials ϕ and A as in (1.4), and E [ Y ] ≡ ∫ Ω Y d P denotes the expected value of the random variable Y . Note that unlike S in (4.4), the actionfunctional ¯ S is not a random variable, as the dependence on ω ∈ Ω is integrated out with respectto the probability measure by calculating the expected value. In fact, S could be regarded as aMonte Carlo approximation of ¯ S when the processes X , . . . , X N are independent and identicallydistributed as X , and similarly for V and P . Similar to the calculations in Section 4.2, the variationsof ¯ S with respect to X , V , and P are given by, respectively, δ X ¯ S = N tot ⋅ E ⎡⎢⎢⎢⎢⎣ P ( T ) ⋅ δ X ( T ) − P ( ) ⋅ δ X ( )⎤⎥⎥⎥⎥⎦ + N tot ⋅ E ⎡⎢⎢⎢⎢⎣ ∫ T ( − ˙P ⋅ δ X − q ∇ x ϕ ( X , t ) ⋅ δ X + q ∑ i,j = V j ∂A j ∂x i ( X , t ) δX i ) dt ⎤⎥⎥⎥⎥⎦ , (5.3) δ V ¯ S = N tot ⋅ E [ ∫ T ( m V + q A ( X , t ) − P ) ⋅ δ V dt ] , (5.4) δ P ¯ S = N tot ⋅ E [ ∫ T ( ˙X − V ) ⋅ δ P dt ] . (5.5)15he variations with respect to A and ϕ are the same as in (4.12) and (4.14), respectively, only withthe charge and electric current densities given by (2.4) rather than (3.2). The total variation of ¯ S with respect to the variations of all arguments is given by δ ¯ S = δ X ¯ S + δ V ¯ S + δ P ¯ S + δ ϕ ¯ S + δ A ¯ S. (5.6) In the following theorems we establish a variational principle for the system of equations (1.2), (2.3),and (2.4) in the special case when g ν ≡ for all ν = , . . . , M . Theorem 5.1 ( Stochastic Lagrange-d’Alembert principle for the VM equations).
Let X , V , P ∈ C ,T be stochastic processes, and let A ∈ X ( R ) , ϕ ∈ X ( R ) be functions. Assume that G ( ⋅ , ⋅ ; f ) is a C function of its arguments, where f is given by (2.4a) . Then X , V , P , A , and ϕ almost surely satisfy the system of random differential equations ˙X = V , (5.7a) P = m V + q A ( X , t ) , (5.7b) ˙ P i = − q ∂ϕ∂x i ( X , t ) + q ∑ j = V j ∂A j ∂x i ( X , t ) + m G i ( X , V ; f ) , (5.7c) for i = , , , together with the Maxwell equations (1.2) , (1.4) , (2.4) on the time interval [ , T ] , ifand only if they satisfy the following variational principle δ ¯ S + mN tot ⋅ E [ ∫ T G ( X , V ; f ) ⋅ δ X dt ] = (5.8) for arbitrary variations δ X , δ V , δ P ∈ C ,T , δ A ∈ X ( R ) , and δϕ ∈ X ( R ) , with δ X ( ) = δ X ( T ) = almost surely, and δ A ( x , ) = δ A ( x , T ) = for all x ∈ R , where the action functional ¯ S is givenby (5.2) .Proof. Similar to the proof of Theorem 4.2, the equations δ ϕ ¯ S = and δ A ¯ S = are equivalent to(1.2a) and (1.2d), respectively. Note that C ,T is a subspace of L ( Ω × [ , T ] , R ) , and ⟨ Y , Z ⟩ = E [ ∫ T Y ⋅ Z dt ] is an inner product on that space. Therefore, by substituting Equations (5.3),(5.4), and (5.5) in Equation (5.8), and using the fact that the variations are arbitrary, we establishequivalence with Equations (5.7a)-(5.7c). Theorem 5.2.
Equation (2.3) with g ν ≡ for ν = , . . . , M and Equation (5.7) are equivalent.Proof. Note that for g ν ≡ the diffusion term in (2.3b) vanishes, meaning that the stochastic process V is almost surely of class C . Similar to the proof of Theorem 4.3, by differentiating both sidesof Equation (5.7b) with respect to time and comparing with Equation (5.7c), one eliminates P andobtains Equation (2.3b). 16 emark. By comparing Theorem 4.2 and Theorem 5.1, one could intuitively expect that anextension of the variational principle (5.8) valid also for nonzero g ν should read δ ¯ S + mN tot ⋅ E [ ∫ T G ( X , V ; f ) ⋅ δ X dt + M ∑ ν = ∫ T g ν ( X , V ; f ) ⋅ δ X ○ dW ν ( t )] = . (5.9)It is indeed easy to verify that if X , V , P , ϕ , and A satisfy (1.2), (1.4), (2.4), then they also satisfy(5.9). The converse statement, however, appears not to be true. A counterexample is provided by,for instance, the Lenard-Bernstein collision operator (see Section 2.2.1). For the forcing terms g ν as in Equation (2.6) the Stratonovich integral terms in (5.9) take the form ∫ T g ν ( X , V ; f ) ⋅ δ X ○ dW ν ( t ) = √ ν c γ ∫ T δX ν ○ dW ν ( t ) (5.10)for ν = , , . The variations δX ν are almost surely of class C , and therefore have sample pathsof almost surely finite variation. Consequently, the quadratic covariation [ δX ν , W ν ] T = almostsurely (see [68]). Since the expected value of the Itô integral with respect to the Wiener process iszero, we altogether have that E [ ∫ T δX ν ○ dW ν ( t )] = . This means that all information about theforcing terms g ν is lost in (5.9), and from such a variational principle alone one cannot concludethat the processes X and V satisfy the stochastic differential system (2.3). Some extra conditionsor a more general form of (5.9) have to be considered in order to include the effects of the forcingterms g ν . In the full Vlasov-Maxwell system the scalar ϕ and vector A potentials are independent dynamicvariables, and as such have to appear explicitly in the action functional alongside the stochasticprocesses X , V , and P . In order to ensure the correct coupling between the stochastic processesand the electromagnetic field, an expected value was necessary in the definition of the action func-tional (5.2). This created a difficulty in deriving a variational principle, as pointed out in the remarkfollowing Theorem 5.2. This difficulty can be circumvented for the Vlasov-Poisson equations be-cause in this case the electrostatic potential ϕ is uniquely determined by the stochastic process X ,as will be demonstrated below. The collisional Vlasov-Poisson equations ∂f∂t + v ⋅ ∇ x f + qm E ⋅ ∇ v f = C [ f ] , (6.1)where E = −∇ x ϕ, (6.2a) ∆ x ϕ = − ρ, (6.2b)17nd the charge density ρ is given by (1.3), are an approximation of the Vlasov-Maxwell equationsin the nonrelativistic zero-magnetic field limit. The associated stochastic differential equations takethe form d X = V dt, (6.3a) d V = ( qm E ( X , t ) + G ( X , V ; f )) dt + M ∑ ν = g ν ( X , V ; f ) ○ dW ν ( t ) . (6.3b)The equations (2.4b), (6.2), and (6.3) form a stochastic reformulation of the Vlasov-Poisson equa-tions. A stochastic particle discretization and the corresponding stochastic variational principle canbe derived just like in Sections 3 and 4, respectively. Also, a variational principle analogous to theLagrange-d’Alembert principle presented in Section 5 can be derived in a similar fashion. However,by doing so, one encounters the same difficulty with including the Stratonovich integral. In the caseof the Vlasov-Poisson equations a different variational principle can be obtained by observing thatthe electrostatic potential ϕ can be expressed as a functional of the stochastic process X , ϕ ∶ R × R × C ,T Ð→ R , (6.4)by solving Poisson’s equation (6.2b). Given the charge density function (2.4b) and specific boundaryconditions, the solution of Poisson’s equation can be written using an appropriate Green’s functionfor the Laplacian. Assuming the spatial domain is unbounded, the standard Green’s function yields ϕ ( x , t, X ) = π ∫ R ρ ( y , t )∣ x − y ∣ d y = qN tot π E [ ∣ x − X ( t )∣ ] . (6.5)From (6.2a) we have the electric field E ( x , t, X ) = qN tot π E [ x − X ( t )∣ x − X ( t )∣ ] . (6.6) Let us consider the action functional ˆ S ∶ Ω × C ,T × C ,T × C ,T × C ,T Ð→ R (6.7)defined by the formula ˆ S [ X , Y , V , P ] = ∫ T ( m ∣ V ( t )∣ − qϕ ( X ( t ) , t, Y ) + P ( t ) ⋅ ( ˙X ( t ) − V ( t ))) dt, (6.8)where the electrostatic potential ϕ is given by (6.5). Note that similar to S in (4.4), the functional ˆ S is itself random, and can be viewed as the action functional of particles represented by the process X which are moving in the electric field generated by particles represented by the process Y . Similarto the calculations in Section 4.2, the variations of ˆ S with respect to X , V , and P are given by,respectively, 18 X ˆ S [ X , Y , V , P ] = P ( T ) ⋅ δ X ( T ) − P ( ) ⋅ δ X ( ) − ∫ T δ X ( t ) ○ d P ( t ) + ∫ T q E ( X ( t ) , t, Y ) ⋅ δ X ( t ) dt, (6.9) δ V ˆ S [ X , Y , V , P ] = ∫ T ( m V ( t ) − P ( t )) ⋅ δ V ( t ) dt, (6.10) δ P ˆ S [ X , Y , V , P ] = ∫ T ( ˙X ( t ) − V ( t )) ⋅ δ P ( t ) dt, (6.11)where the electric field E is given by (6.6). Note that we are not considering variations with respectto Y . Let us for convenience define the joint variation of ˆ S with respect to X , V , and P as δ ( X , V , P ) ˆ S = δ X ˆ S + δ V ˆ S + δ P ˆ S. (6.12) In the following theorem we formulate a variational principle for the system of equations (2.4b),(6.2), and (6.3). Note that E ( X ( t ) , t, X ) is the electric field generated by a distribution of chargedparticles represented by the process X at time t , and evaluated at the random point x = X ( t ) inspace. Furthermore, the notation δ X ˆ S [ X , X , V , P ] means that the variation of ˆ S is evaluated forthe arguments X , Y , V , P with Y = X . Theorem 6.1 ( Stochastic Lagrange-d’Alembert principle for the VP equations).
Let X ∈ C ,T and V , P ∈ C ,T be stochastic processes, and let ϕ ( ⋅ , ⋅ , X ) ∈ X ( R ) be given by (6.5) .Assume that G ( ⋅ , ⋅ ; f ) and g ν ( ⋅ , ⋅ ; f ) for ν = , . . . , M are C functions of their arguments, where f is given by (2.4a) . Then X , V , and P satisfy the system of stochastic differential equations ˙X ( t ) = V ( t ) , (6.13a) P ( t ) = m V ( t ) , (6.13b) d P ( t ) = ( q E ( X ( t ) , t, X ) + m G ( X ( t ) , V ( t ) ; f )) dt + m M ∑ ν = g ν ( X ( t ) , V ( t ) ; f ) ○ dW ν ( t ) , (6.13c) on the time interval [ , T ] , if and only if they satisfy the following variational principle δ ( X , V , P ) ˆ S [ X , X , V , P ] + m ∫ T G ( X , V ; f ) ⋅ δ X dt + m M ∑ ν = ∫ T g ν ( X , V ; f ) ⋅ δ X ○ dW ν ( t ) = (6.14) for arbitrary variations δ X ∈ C ,T , and δ V , δ P ∈ C ,T , with δ X ( ) = δ X ( T ) = almost surely,where the action functional ˆ S is given by (6.8) .Proof. Analogous to the proof of Theorem 4.2. 19 emark.
It is straightforward to see that Equations (6.13), together with (6.5) and (6.6), areequivalent to the system of equations (2.4b), (6.2), and (6.3). The Lagrange-d’Alembert principle(6.14) is unusual in that the variations of the action functional ˆ S with respect to the argument Y are omitted. Thanks to such a form, however, the action functional does not require an expectedvalue, and the collisional effects can be correctly included. A similar idea to solve Poisson’s equationand plug the solution into the action functional was presented in [90], where the authors proposed avariational principle for the collisionless Vlasov-Poisson equations. In that approach the energy ofthe electric field was also included in the variational principle, and the variations were taken withrespect to all arguments of the action functional. This approach could be adapted to the stochasticreformulation of the Vlasov-Poisson equations, but the corresponding action functional would havea form similar to (5.2), that is, it would need to contain an expected value, and therefore we wouldface a similar difficulty as for the Vlasov-Maxwell equations in Section 5.2. In this work we have considered novel stochastic formulations of the collisional Vlasov-Maxwell andVlasov-Poisson equations, and we have identified new stochastic variational principles underlyingthese formulations. We have also proposed a stochastic particle method for the Vlasov-Maxwellequations and proved the corresponding stochastic variational principle.Our work can be extended in several ways. The stochastic variational principle introduced inSection 4 can be used to construct stochastic variational particle-in-cell numerical algorithms for thecollisional Vlasov-Maxwell and Vlasov-Poisson equations. Variational integrators are an importantclass of geometric integrators. This type of numerical schemes is based on discrete variationalprinciples and provides a natural framework for the discretization of Lagrangian systems, includingforced, dissipative, or constrained ones. These methods have the advantage that they are symplecticwhen applied to systems without forcing, and in the presence of a symmetry, they satisfy a discreteversion of Noether’s theorem. For this reason they demonstrate superior performance in long-timesimulations; see [30], [40], [41], [53], [54], [61], [64], [65], [71], [81], [82]. Variational integrators wereintroduced in the context of finite-dimensional mechanical systems, but were later generalized toLagrangian field theories (see [60]) and applied in many computations, for example in elasticity,electrodynamics, fluid dynamics, or plasma physics; see [45], [55], [67], [75], [77], [80], [88], [89].Stochastic variational integrators were first introduced in [9] and further studied in [8], [36], [38],[48], [83].It would also be interesting to determine how to modify the variational principle (5.8), or whatadditional conditions are necessary in order to extend Theorem 5.1 to nonzero forcing terms g ν .Another aspect worth a more detailed investigation is the issue of existence and uniqueness ofthe solutions of the stochastic reformulations presented in this work, which are nontrivial systemsof coupled stochastic and partial differential equations. This question is closely connected to theissue of existence and uniqueness of the solutions of the original collisional Vlasov-Maxwell and theVlasov-Poisson equations. General results are available in the collisionless case (see, e.g., [20], [86],[87]), but the theory for the collisional equations is less developed (see [19], [22], [63], [66], [78], [84],[85] and the references therein). 20 cknowledgements We would like to thank Christopher Albert, Darryl Holm, Michael Kraus, Omar Maj, HoumanOwhadi, Eric Sonnendrücker, and Cesare Tronci for useful comments and references. The study is acontribution to the Reduced Complexity Models grant number ZT-I-0010 funded by the HelmholtzAssociation of German Research Centers.
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