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Formation of Singularities in Madelung Fluid: a Nonconventional

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Formation of Singularities in Madelung Fluid: a Nonconventional Formation of singularities in Madelung fluid: a nonconventional application of Itˆocalculus to foundations of Quantum Mechanics Laura M. Morato1 Facolt`adi Scienze, Universit`adi Verona Strada le Grazie, 37134 Verona, Italy [email protected] Summary. Stochastic Quantization is a procedure which provides the equation of motion of a Quantum System starting from its classical description and incorpo- rating quantum effects into a stochastic kinematics. After the pioneering work by E.Nelson in 1966 the method has been developed in the eighties in various differ- ent ways. In this communication I summarize and systematize the results obtained within an approach based on a Lagrangian variational principle where 3/2 order contributions in Itˆocalculus are required, leading to a generalization of Madelung fluid equations where velocity fields with vorticity are allowed. Such a vorticity induces dissipation of the energy so that the irrotational solu- tions, corresponding to the usual conservative solutions of Schroedinger equation, act as an attracting set. Recent numerical experiments show generation of zeroes of the density with concentration of vorticity and formation of isolated central vortex lines. 1 Introduction This communication is concerned with an application of Itˆocalculus to the problem of describing the dynamical evolution of a quantum system once its classical description (which can be given in terms of forces, lagrangian or hamiltonian) is given. We know that, if the classical hamiltonian is given, the canonical quantization rules lead to Schr¨odinger equation, which beautifully describes the behavior of microscopical systems. But we also know that this procedure seems to fail when applied to microscopical systems interacting with a (macroscopic) measuring apparatus. This fact has been a motivation for investigating other quantization procedures. In his pioneering work in 1966 E. Nelson proposed a Stochastic Quantiza- tion (often called Stochastic Mechanics) where, given the forces acting on the system, quantum effects are incorporated into a stochastic kinematics [18]. This approach was widely developed during the eighties, with the introduc- tion of stochastic variational principles (see for example [19], [2] , [15] and 2 Laura M. Morato references quoted therein). I present here a synthesis of the results obtained within an approach which leads to a dissipative generalization of Schr¨odinger equation, the usual conservative solutions being in fact dynamical equilibrium states which form an attracting set [13] [14][10]. The basic tool is Itˆocalculus 3 where stochastic increments must be estimated to the order 2 . For a quantum particle of mass m, subjected to a force which is the gra- dient of a scalar potential Φ, Schr¨odinger equation reads 1 ı ∂ Ψ = − 2∇2 + Φ Ψ (1) ~ t 2m~ ψ denoting the quantum mechanical wave function. By a change of variables Schr¨odinger equation can be formally written in a fluidodynamical version, the so called Madelung fluid equations . ( ∂tρ = −∇ · (ρv) √ 2 ∇2 ρ (2) ~ √ 1 ∂tv + (v · ∇) v − 2m2 ∇ ρ = − m ∇Φ where ρ = |ψ|2 v = ∇S S being the phase of the wave function ψ. The equivalence is only formal if the density ρ is not strictly positive at all times. The velocity field of Madelung fluid is irrotational in all points where the density is different from zero. In many examples solutions of Schr¨odinger equa- tion which exhibit nodes correspond to solutions of Madelung fluid equations with singular velocity and isolated vortex lines. In our setting we are led to a dissipative generalization of such equations, which allow velocity fields with a distributed vorticity. It was conjectured that such a vorticity asymptotically can concentrate in the zeroes of the den- sity, describing the formation of the singularities and in particular of isolated vortex lines. The problem is very difficult from the analytical point of view but recent numerical results seem to confirm this conjecture [3]. It is worth stressing that arrays of isolated vortex lines are observed in quantum fluids, as liquid Helium and Bose Einstein condensates (see [9], [11], [12], [1]), but the mechanism underlying their formation is still not well understood. Describing the formation of isolated vortex lines in Madelung fluid, from smooth initial data, could represent a contribution to the solution of this problem. Formation of singularities in Madelung fluid 3 2 A stochastic quantization procedure For a quantum particle of mass m in a scalar potential Φ we denote its config- uration at time t by q(t). We model the evolution in time of the configuration by a “smooth diffusion”, in the following sense: Definition 1. A diffusion q is a “smooth diffusion” if 1) Its drift v+ is a smooth (i.e. infinitely differentiable) time dependent ~ vector field and its diffusion coefficient is constant (in this setting equal to m , ~ denoting Planck’s constant divided by 2π) 2) There exists a probability space (Ω, F,P ) and a standard Brownian Motion W s.t., for t ∈ [0,T ], T > 0, 1 Z t 2 ~ q(t) = q(0) + v+ (q(s), s) ds + W (t) (3) 0 m 3) There exists a reversed standard Brownian Motion W ∗ on (Ω, F,P ) and v− s.t., for any t ∈ [0,T ], 1 Z t 2 ~ ∗ ∗ q(t) = q(0) + v− (q(s), s) ds + (W (t) − W (0)) (4) 0 m I recall that a reversed standard Brownian Motion W ∗ on the finite time interval [0.T ] is defined by the equality W ∗(t) = Wˆ (T − t), t ∈ [0,T ] (5) Wˆ still denoting a standard Brownian Motion. The finite energy condition is sufficient for property 3) (See [5]. An exten- sion to the infinite dimensional case is given in [6]). We also recall that if ρ is the (time dependent) density of a smooth diffusion one has, in particular v − v + − = ~ ∇ ln ρ (6) 2 2m ∂tρ = −∇ · (ρv) (7) were v is the “current velocity”, defined as v + v v := + − (8) 2 For any finite time interval [ta, tb] and positive integer N we fix the nota- tions tb−ta ∆ := N + ∆ q(ti) := q(ti+1) − q(ti) future increment − ∆ q(ti) := q(ti) − q(ti−1) past increment 4 Laura M. Morato We now consider the following mean discretized version of the classical action functional N + + N X 1 ∆ q(ti) · ∆ q(ti) A [q] := E m − Φ(q(ti)) ∆ (9) [ta,tb] 2 ∆2 i=1 were q is uniquely determined by the triple [W, v+, qo] and E denotes the expectation. + By exploiting the backward representation and estimating ∆ q(ti) to the 3 order ∆ 2 , which gives 1 2 ∆+q(t) = ~ ∆+W (t) + v (q(t), t) ∆+ m + 1 3 " # 2 X Z t+∆ + ~ ∂ v (q(t), t) (W (s) − W (t)) ds + m k + k k k=1 t 3 + o(∆ 2 ) (10) we find N + − X 1 ∆ q(ti) · ∆ q(ti) AN [q] = E m [ta,tb] 2 ∆2 i=1 3 + ~ + o(∆) − Φ(q(t )) ∆ (11) 2 ∆ i In order to generalize the classical action principle, starting from the above defined functional, two methods have been considered. The former, that will be called Eulerian or Stochastic Control approach, consists in eliminating the divergent term in the discretized action and then take the limit for N going to infinity. After simple manipulations one can see that such a limit can be expressed as a simple functional of the drift field v+. This allows to exploit stochastic control like techniques [8]. The latter, that will be called Lagrangian or path-wise approach, consists in taking pathwise variations of q for fixed W in AN [q]. This eliminates the divergent term .The limit for N going to [ta,tb] infinity is taken only at the end of the calculus of variations (see [13], [14] and [10]).This is the approach considered in the following. Definition 2. The set of admissible test diffusions for a given W is consti- tuted by the set of all smooth diffusions associated to W according to the previous definition. For the test diffusion q(t) at time t let q0(t) := q(t) + δq(t) denote the varied diffusion. We require that this is still a smooth diffusion with the same 0 W . Therefore there must exist a smooth drift field v+ such that Formation of singularities in Madelung fluid 5 1 Z t 2 ~ q(t) = q(0) + v+ (q(s), s) ds + W (t) (12) 0 m 1 Z t 2 0 0 0 ~ q (t) = q(0) + v+ (q (s), s) ds + W (t) (13) 0 m We introduce the variation process h and the variation of the drift f by putting, for > 0, ( εh(t) := δq(t) ε > 0 0 (14) εf := v+ − v+ Then one finds 3 ˙ X h(t) = ∂jv+ (q(t), t) hj(t) + f (q(t), t) (15) j=1 so that h(t) is a differentiable stochastic process. It satisfies a first order ODE for every realization of q. As a consequence h cannot be fixed both in ta and tb. This fact, which has no counterpart in the classical case, comes to be a typical quantum peculiarity. Definition 3. A process h will be said “admissible variation” for the test diffusion q if it is solution of (15) for a smooth f. We want now to characterize the motions which are represented by “critical diffusions” : Definition 4. A smooth diffusion q∗ is critical with fixed initial position if, ∀ h admissible, n N ∗ N ∗ o lim A[t ,t ] [q + εh] − A[t ,t ] [q ] − εptb htb = o(ε) (16) N↑∞ a b a b ∗ h(ta) = 0 and a smooth diffusion q is critical with fixed final position if, ∀ h admissible, n N ∗ N ∗ o lim A[t ,t ] [q + εh] − A[t ,t ] [q ] + εpta hta = o(ε) (17) N↑∞ a b a b h(tb) = 0 pta and ptb are fixed random variables playing the role of the classical initial and final “momentum”.
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