Stochastic Theories and Deterministic Differential Equations

Volume 2010, Article ID 749306,29pages doi:10.1155/2010/749306

Research Article

Stochastic Theories and Deterministic Differential Equations

John F. Moxnes

and Kjell Hausken

1Department for Protection, Norwegian Defence Research Establishment, P.O. Box 25, 2007 Kjeller, Norway

2Faculty of Social Sciences, University of Stavanger, 4036 Stavanger, Norway

Correspondence should be addressed to John F. Moxnes,john-f.moxnes@ffi.noand Kjell Hausken,kjell.hausken@uis.no

Received 3 January 2010; Accepted 6 May 2010 Academic Editor: Nakao Hayashi

Copyrightq2010 J. F. Moxnes and K. Hausken. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We discuss the concept of “hydrodynamic” stochastic theory, which is not based on the traditional Markovian concept. A Wigner function developed for friction is used for the study of operators in quantum physics, and for the construction of a quantum equation with friction. We compare this theory with the quantum theory, the Liouville process, and the Ornstein-Uhlenbeck process.

Analytical and numerical examples are presented and compared.

1. Introduction

Stochastic theories model systems which develop in time and space in accordance with probabilistic laws.The space is not necessarily the familiar Euclidean space for everyday life. We distinguish between cases which are discrete and continuous in time or space. See Doob 1 or Taylor and Karlin 2 for a mathematical definition of stochastic processes, which is not replicated here. Briefly, the usual situation is to have a set of random variables {Xt} defined for all values of the real number t say time, which could be discrete or continuous. The outcome of a random variable is a state value often a real number. The set of random variables are called a stochastic process, which is completely determined if the joint distribution of the set of random variables{Xt}is known. A realization of the stochastic process is an assignment to eachtin the set{Xt},a value ofX_t.Essential in stochastic theories is how randomness is accounted for. For Markov 3 processesin the narrowest sense, a stochastic process has the Markov property if the probability of having stateX_tΔtat timeth, conditioned on having the particular statex_tat time t, is equal to the conditional probability

of having that same stateX_tΔtbut conditioned on its value for all previous times beforet.

See Feller4for a broader definition., which are an important class of stochastic processes, a recurrence relation is used such that the state value ofX_tΔtat timet Δtis given by the state value at timet, plus a state value of a random variable at time t.By “counting up”

the different realizationstracksin the state space the joint distribution can be constructed.

Although counting up all different realizations in general constructs the joint probability, the inverse does not hold. Hence the joint probability of the set of random variables {Xt} does not lead to a unique recurrence relation.A random “disturbance” in a Markov process may possibly influence all subsequent values of the realization. The influence may decrease rapidly as the time point moves into the future.A Markov process may be deterministic, that is, all values of the process at timet > t are determined when the value is given at timet. Or a process may be nondeterministic, that is, a knowledge of the process at timet is only probabilistically useful in specifying the process at timet > t.This paper considers a so-called “hydrodynamic” approach to account for randomness. We specify constitutive relations in an equation set akin to what is used in hydrodynamic formulations of gas flow e.g.,5.

Consider the variables as position and velocity for illustration, but the method applies generally. By integrating the equation for the joint distribution for two stochastic variables with respect to the second variablevelocity, the well-known equation for the conservation of probability in space is found. This equation, which is only the conservation of probability, can be used without referring to any stochastic theory. The equation includes the so-called current velocity. It is well known that in Boltzmann kinetic theory or in most Langevin models, the total derivative of the current velocity is equal to the classical force minus a term that is proportional to 1/ρ_Xx, ttimes the space derivative of VarYt/X_tρXx, t, where VarYt/X_tis the variance ofY_tat timetgiven the position ofX_t,andρ_Xx, tis the density of Xt 6. Now, the equation for the conservation of probability in space is a first partial differential equation. As a second equation, set the total derivative of the current velocity equal to the classical force minus a term that is proportional to 1/ρ_Xx, t times the space derivative of−ρXx, tVarYt/Xtas in Boltzmann’s kinetic theory or in Langevin models.

Thus randomness can be accounted for by constitutive relations for VarYt/Xt without postulating a relation for a joint or quasi joint distributionLarsen 1978,5. For the Liouville process realizations in the position- velocity spacephase spacecannot cross. In addition, for a conservative classical force, all realizations that start at the same position will have a unique velocity at a given position when applying the Liouville process, which implies VarYt/Xt 0. The equation for the total derivative of the current velocity, which now equals the classical force, can be integrated in space to give the familiar Hamilton-Jacobi equation in classical mechanics as a special case. More generally, the total derivative of the current velocity of the Liouville and the Ornstein-Uhlenbeck7processesassuming uncorrelated Gaussian noise has also been analyzed when assuming initial conditions in position and velocity that are independent and Gaussian distributed. It has been shown that VarYt/X_tis independent ofx, but time dependent for the free particle or for the harmonic oscillator8–

10. We believe that this hydrodynamic method can be useful when experimental data pertain to the variables in the equation set, and there is no direct experimental access to microscopic dynamics.

We follow the idea that stochastic processes could in some way be used to understand quantum mechanics 11–13 Kaniadakis 14 by studying the Liouville process and the Ornstein-Uhlenbeck process more carefully in relation to the well-known and so-called

“operator ordering problem” in quantum physics 15–21. A relation for the operator

ordering ofp²q² is important for quantization of the kinetic energy in a curved space. We use an approach different from that usually presented in the literaturesee also Moxnes and Hausken22.

Quantum mechanics based on the Schr ¨odinger equation makes it difficult to describe irreversible processes like the decay of unstable particles and measurements processes.

The fact that classical and quantum systems must be coupled by a dissipative rather than reversible dynamics follows from the no-go theorem, where it was shown in a general framework that the information of the measured object cannot be transmitted to values of macroscopic observables as long as the dynamics of the total system is reversible in time 23,24. seeAppendix B. See also Bell25, Bell 26, Haag 27, Blanchard and Jadczyk 28, Haag29, Machida and Namiki30, Araki31, Araki32, Ozawa24, Olkiewicz 33for the literature related to the measuring problem.

Macroscopic systems are usually described either by classical physics of a few classical parameters or by quantum statistical mechanics if the quantum nature is essential. But in the Machida-Namiki model of measurement a new formulation of describing the process of measurement is given. The measured object is microscopic but the measuring apparatus is described macroscopically24,30–32. However, there are examples of macroscopic quantum phenomena where a large number of particles can be described by a few degrees of freedom.

In these cases the evolution of the quantum object depends on the classical environment, but also a modification of the dynamics of the classical system through some expectations values appears. Friction arises from the transfer of collective translational kinetic energy into nearly random motion and can formally be considered as resulting from the process of eliminating the microscopic degrees of freedom. This paper does not review the various attempts to solve the difficulties associated with the measuring problem, but we observe that one of the many attempts to overcome the difficulties has led to the development of the so- called collapse theories, that is, to the dynamical reduction program34. This theory accepts a modification of the standard evolution law such that micro processes and macro processes are governed by a unique dynamics. The dynamics implies that the micro-macro interaction in a measurement process leads to the wave packed reduction.See Giancarlo Chirardi2007 athttp://www.plato.stanford.edu/entries/qm-collapse, for review of collapse theories and Efinger35for a nonlinear unitary framework for quantum state reduction. See Bassi et al.

36for experiments that could be crucial to check the dynamical reduction models versus quantum mechanics.

A linear friction term together with an uncorrelated random Gaussian noise term is inherent in most classical Langevin models. The Boltzmann distribution is achieved as a steady-state solution. Notice that when friction and the random term are zero, every solution for the joint density of the typeρt, x, y ρHis a steady-state solution of the Liouville equation, whereHis the Hamiltonian. This shows the importance of linear friction to achieve the Boltzmann distribution as the steady-state asymptotic classical behavior. Interestingly, models accounting for friction have been further developed into so-called quantum Langevin models for quantum noise 37. The transformation to quantum mechanics is pursued by using the Heisenberg picture of quantum mechanics. That is, the transformation to quantum mechanics is achieved by letting position and momentum be transformed to the corresponding operators. In the Heisenberg picture the concept of a Hamiltonian is not generally necessary, and friction can be incorporated. To study the “measuring problem”

this paper does not use the common Heisenberg picture but instead a nonlinear quantum equation accounting for linear friction in the Schr ¨odinger picture. We use a differential equation for the Wigner function including linear friction to establish a quantum equation

accounting for linear friction. The friction describes the interaction with the measuring device 22.

Section 2considers second-order processes and joint distributions.Section 3develops a general stochastic theory not based on joint distributions.Section 4considers the operator problem. Section 5 formulates the quantum equation with friction. Section 6 compares different solutions from the Liouville and the Ornstein-Uhlenbeck processes with different solutions from the quantum equation with and without friction.Section 7concludes.

2. Some Relations That Follow for Phase-Space Functions

In classical physics or in the phase-space formulation of quantum physics, two dimensional systems are generally described by a phase-space distribution or quasi distribution. In this section we study the following equation for the phase-space function:

ρ˙

t, x, y mod −D1

ρ t, x, y

y

−D2

ρ t, x, y

fY

x, y

−βtD1D₂ ρ

t, x, y 1

2!D²₂

g_2Yt, xρ

t, x, y 1

3!D₂³

g3Yt, xρ

t, x, y 1

4!D⁴₂

g4Yt, xρ

t, x, y · · ·, ρ_Xt, x^def

ρ

t, x, y dy, ρ_Y

t, ydef

ρ t, x, y

dx, E

X_tⁿY_t^mdef

ρ t, x, y

xⁿy^mdx dy 2.1 where “def” means definition and “mod” means model assumptions.βtis some function.

It can be shown that βt can be related to correlation 38, 39. However, as it stands in 2.1such an interpretation is not needed. But for simplicity we callβta correlation factor.

g2Yt, x, g3Yt, x, g4Yt, x, . . . are some functions, and ρt, x, y is a so-called quasi joint distribution which does not have to be positive definite for all parametric functions. Examples of special cases of 2.1are the Ornstein-Uhlenbeck process7, the Liouville process, and the differential equation for the Wigner15function as defined by Gardiner and Zoller37, page 126orAppendix Cfor further details. Equation2.1corresponds to adding terms to the Liouville equation that includes higher-order derivative of the momentum velocity.

Integrating2.1gives

˙

ρXt, x D1

ρXt, xvXt, x

0, 2.2a vXt, x^def

ρ t, x, y

ydy

ρ_Xt, x , v2t, x^def ρ

t, x, y y²dy

ρ_Xt, x , 2.2b

˙

vXt, x v_Xt, xD1

ρ_Xt, xvXt, x

ρXt, x 1

ρXt, x

ρ˙ t, x, y

ydy, 2.2c

1 2D₁

v_Xt, x² 1

2D₁ ρ

t, x, y ydy₂ ρ_Xt, x² −vXt, x²D₁ρ_Xt, x

ρ_Xt, x v_Xt, xD1

ρXt, xvXt, x ρ_Xt, x .

2.2d

This gives that 1 ρXt, x

ρ˙

t, x, y

ydy 1

ρXt, x

×

−D1

ρ t, x, y

y

−D2

ρ t, x, y

fY

x, y

−βtD1D2

ρ

t, x, y

D²₂ 2

g2Yρ

t, x, y

· · · y dy

− 1 ρXt, xD₁

ρ

t, x, y

y²dy 1 ρXt, x

ρ

t, x, y f_Y

x, y dy

βt ρ_Xt, xD₁

ρ

t, x, y dy

− 1 ρXt, xD1

ρXt, xv2t, x

fYx βtD1ρXt, x ρXt, x , f_Yx^def

ρ t, x, y

fY

x, y dy ρXt, x ,

2.3

and finally that

˙

ρXt, x D1

ρXt, xvXt, x

0, 2.4a

˙

vXt, x vXt, xD1vXt, x v_Xt, xD1

ρ_Xt, xvXt, x

ρXt, x − 1

ρXt, xD₁

ρ_Xt, xv2t, x f_Yx βtD₁ρ_Xt, x

ρ_Xt, x −v_Xt, x²D₁ρ_Xt, x

ρ_Xt, x vXt, xD1

ρXt, xvXt, x ρ_Xt, x

fYx D1

ρXt, x

v²_Xt, x−v2t, x qβt ρXt, x

f_Yx D1

ρXt, xΠt, x ρ_Xt, x ,

2.4b

Πt, x^def v_Xt, x²−v₂t, x βt, Var Yt

X_t

v_Xt, x²−v₂t, x. 2.4c

Equation 2.4a is the familiar conservation of probability density in space. Notice that if ρt, x, yis a Dirac delta function in y, the termvXt, x² −v₂t, xbecomes zero. This is achieved for the Liouville process and a conservative force if the initial values in position are a Dirac delta function, which means that all realizations start from a common position.

Then the Liouville process can be given a more simplified expression through the well known Hamilton-Jacobi equation.

The equations in2.4a–2.4care not closed due tof_Yxandv₂t, x or alternatively VarYt/Xt. VarYt/Xt can be calculated explicitly for a Gaussian initial distribution by applying the Gaussian uncorrelated noise 9. It is found that the term is independent of x and that for a free particle for the Ornstein-Uhlenbeck7process

− D₁

ρ^OU_X t, xVar^OUYt/X_t ρ^OU_X t, x

4

a²b²2a²qt2

3b²qt³ 1 3q²t⁴

D1

1/

2

ρ_X^OUt, x

D²₁

ρ^OU_X t, x

.

2.5

The initial distribution is in this case chosen as two independent Gaussian distributions.a²is the initial variance in position andb²the variance in velocity.qg2Y/2 is the diffusion coefficient that is now assumed to be constant. The Liouville solution follows as a special case when the diffusion coefficientqis set to zero.

3. The Hydrodynamic Method: A Stochastic Theory Not Based on Joint Distributions

By using 2.1, 2.4a–2.4c follows for the Ornstein-Uhlenbeck7 process, the Liouville process or the quantum theory based on the Wigner function. Generally, Boltzmann kinetics also allows the same mathematical structure6. More generally, we formulate a stochastic theory by constitutive equations in the equation for the total derivative of the current velocity, akin to what is used for hydrodynamic theories, to read

˙

ρXt, x D1

ρXt, xvXt, x

0, 3.1a

˙

vXt, x vXt, xD1vXt, x fx D₁

ρ_Xt, xΠt, x

ρXt, x . 3.1b

The stochastic theory is fully described by postulating a so-called “constitutive” relation for Πt, x. A joint or quasi joint phase-space distribution is not used. It follows from3.1aand 3.1bthat

EX˙ t

˙

ρ_Xt, xx dx−

D₁

ρ_Xt, xvXt, x x dx

ρ_Xt, xvXt, xdx, EX¨ t

∂

∂t

ρ_Xt, xvXt, xdx

˙

ρ_Xt, xvXt, x v˙_Xt, xρXt, x dx

v˙Xt, x vXt, xD1vXt, xρXt, xdx

ρ_Xt, xfx D₁

ρ_Xt, xΠt, x

dxE

fx

.

3.2

Thus we fulfill our crucial equation ¨EXt EfXt. As three test examples, we set the constitutive relation in3.1aand3.1bas

Alt1 :Πt, x ^mod Π^Qt, x^def 1 4ρ^Q_Xt, x

⎛

⎜⎝D²₁ρ_X^Qt, x−

D₁ρ^Q_Xt, x₂ ρ_X^Qt, x

⎞

⎟⎠, 3.3a

⇒ D₁

ρ^Q_Xt, xΠ^Qt, x ρ^Q_Xt, x D1

D₁²ρ^Q_Xt, x^1/2

2ρ^Q_Xt, x^1/2 , 3.3b

Alt2 :Πt, x ^mod Πt, x^{QC def} Π^Qt, x βt, 3.3c

⇒ D₁

ρ^QC_X t, xΠ^QCt, x ρ_Xt, x D1

D₁²ρ^QC_X t, x^1/2

2ρ_X^QCt, x^1/2 βtD₁ρ^QC_X t, x

ρ_X^QCt, x , 3.3d Alt3 :Πt, x^def Π^OUCt, x ^mod

a²b²2a²qt 2/3b²qt³ 1/3q²t⁴ ρ^OUC_X t, x

×

⎛

⎜⎝D²₁ρ_X^OUCt, x−

D₁ρ^OUC_X t, x₂ ρ^OUC_X t, x

⎞

⎟⎠βt

⇒ D₁

ρ^OUC_X t, xΠ^OUCt, x ρ_X^OUCt, x 4

a²b²2a²qt2

3b²qt³1 3q²t⁴

×D1

D²₁ρ^QUC_X t, x^1/2

2ρ_X^QUCt, x^1/2 βtD₁ρ^QUC_X t, x ρ^QUC_X t, x , Alt4 :Πt, x const.K.

3.3e

Alternative 1superscriptQcorresponds to the quantum theory as we will show. Alternative 2superscript QCcorresponds to what we call quantum theory with exponential correlation.

Alternative 3superscript OUCmimics the classical results for the Ornstein-Uhlenbeck7 process. Correlation is included through the correlation factorβt 38,39. As an example, alternative 4 gives an ideal fluid whereKρXt, xis the “pressure”.

We now show the well-known results that the constitutive equation3.3aleads to the Schr ¨odinger equation. First we set that the current velocity is a gradient, to readv^Q_Xt, x D1St, x. This gives after one integration of3.1ba kind of “Hamilton-Jacobi type system”, to read

∂ρ^Q_Xt, x

∂t D₁

ρ^Q_Xt, xv^Q_Xt, x

0, 3.4a

∂St, x

∂t 1

2D1St, x²Vx

D₁²ρ^Q_Xt, x^1/2

2ρ^Q_Xt, x^1/2 . 3.4b

We use the traditional mathematical trick and introduce the well-known Madelung decom- position, to readψt, x^def ρ_X^Qt, x^1/2Expi St, x. This allows the two nonlinear equations in3.4aand3.4bto be written as one linear equation for the in general complexψt, x, to read12

−1

2D₁²ψt, x Vxψt, x iψt, x˙ 3.5

which is the Schr ¨odinger equation.We use units such that the massm1 and the reduced Planck constant1.For alternative 2 we achieve that

−1

2D₁²ψt, x Vxψt, x−βtLn

λψt, x^∗ψt, x

iψt, x.˙ 3.6

The constantλis arbitrary. Equation3.6is a kind of Schr ¨odinger equation accounting for correlation. Without correlationβt 0. For alternative 3 we achieve

−1

2h²tD₁²ψt, x Vxψt, x−βtLn

λψt, x^∗ψt, x

iψt, x,˙ h²t 4

a²b²2a²qt2

3b²qt³1 3q²t⁴

.

3.7

Notice that whenβt 0 anda²b² 1/4 we achieve the Schr ¨odinger equation. In ordinary unitsa²b² 1/4 means that VarXt0VarYt0 1/4²/m. Alternative 4 simply gives3.6 without theD²₁term.

4. Operators

When the equation set3.1aand3.1bis postulated together with a constitutive equation forΠt, xfor a stochastic theory, the equation set does not show a way of calculatingEYtn, simply because there is, as such, no stochastic variable Yt velocity or momentumin the theory. But this has caused problems in quantum physics. Briefly, Following the traditional concept in quantum physics, we write

θ p, t_def

1 2π^1/2

ψ

t, q Exp

−iqp dq,

ψ

t, q_∗ ψ

t, q

dq1, δ q_def

1 2π

Exp

ipq dq,

E p^mdef

p^mθ

p, t_∗ θ

p, t dp.

4.1

It follows that

θ p, t_∗

θ p, t

dp 1

2π

ψ t, q_∗

Exp ip

q−q ψ

t, q

dp dq dq

ψ t, q_∗

δ q−q

ψ t, q

dq dq

ψ t, q_∗

ψ t, q

dq1.

4.2

Further we achieve that

E p^mdef

p^mθ

p, t_∗ θ

p, t dp

1 2π

p^mψ

t, q_∗ Exp

ip q−q

ψ t, q

dp dq dq

ψ t, q_∗

−i ∂

∂p m

Exp ip

q−q ψ

t, q

dp dq dq

ψ t, q_∗

−iD1^mψ t, q

dq

ψ t, q_∗

p_opm

ψ t, q

dq,

E q^m

θ t, p_∗

iD1^mθ t, p

dpE q^m

θ t, p_∗

qop

_m θ

t, p dp,

4.3

where thepoperator raised to the power ofmbecomesp^m_op^def −iD1^m −i∂/∂q^m. We use the definition

Iα iα xψ

t, q

−E q

ψ t, q

1

i

D₁ψt, q−E p

ψ t, q

2

dq≥0. 4.4

Expanding the terms in 4.4 gives Varp ≥ −α1 αVarq. The right-hand side has a maximum for α −1/2 Varq. This gives VarqVarp ≥ 1/4, which is the uncertainty relation. However, these equations above in this section are well known, purely mathematical and follow as seen directly from the definitions. No physics is involved so far. So why is p associated with velocity or momentum? Say that we calculate∂/∂t EXt. By using the Madelung decomposition we set for any stochastic theory v_Xt, x D₁1/2iLnψt, x/ψ^∗t, x. We find when using the conservation of probability

in space that EX˙ t

∂

∂t

ρ_Xt, xx dx

˙

ρ_Xt, xx dx

x

−D1

ρ_Xt, xvXt, x dx

ψt, xψ^∗t, xD11 2iLn

ψt, x ψ^∗t, x

dx

ψ^∗−iD1ψ dx.

4.5

Thus we find quite generally that ∂/∂tEXt Ep. Then the next question is why is

∂/∂tEXtassociated with the expectation of velocity, or more generallyEYtm → Ep^m? The arrow means association in the sense that the right-hand side of the arrow is associated to mean the same as the left-hand side. The velocity is a classical concept when “trajectories” are differentiable. Notice that the formulation of quantum mechanical operators from its classical counterpart is straightforward as long as the classical quantity is either a function ofxqor yp, or if it is the sum of such functions. One merely replacespby the operator ofpandq by the operator ofq. But if the classical counterpart contains product terms ofqandp, then difficulties arise, because the received quantum theory gives no unique way of forming the quantum mechanical operator. The phase-space formulation of quantum mechanics gives a solution to the operator problem. Once a quasi phase-space probability density is chosen, each such function would lead to a unique operator ordering, and any chosen operator ordering forqⁿp^m leads to a unique quasi probability distribution 15–21. These quantum quasi distributions have been widely used in quantum optics and in optical image processing 40–42. The Margenau-Hill quasi probability density distribution is equivalent to the rule of symmetrization. Thus logically, there are an infinite number of quantum theories, one for each chosen phase-space quasi joint distribution. However, interestingly, for the special case qpa unique operator seems to be given. That means that the operator is independent of the chosen type of quasi joint distribution. This operator is found by the associationEXtY_t → 1/2∂EX²_t/∂t. In general an association isEXⁿ_tYt → ∂/∂t1/n1EX_tⁿ¹. Thus the operator ofxⁿyis assumed to be found by simply calculating∂/∂t1/n1EXtn1. This is not only associations but actually mathematical deductions if we use a joint or quasi joint distribution of the type in2.1, to read

E˙ Xⁿ_t

∂

∂t

ρ t, x, y

xⁿdx dy

ρ˙ t, x, y

xⁿdx dy

⎛

⎜⎜

⎝

−D1

ρ t, x, y

y

−D₂ ρ

t, x, y f_Y

x, y

−βtD1D₂ ρ

t, x, y

1 2

D₂²

ρ t, x, y

g_2Y x, y

· · ·

⎞

⎟⎟

⎠xⁿdx dy

nE Xⁿ⁻¹_t Yt

.

4.6 Thus the Liouville process and the Ornstein-Uhlenbeck7process allow taking the time derivative inside the expectation. But notice that the results are based on a specific joint distribution or a quasi joint distribution of the type in2.1. So∂/∂t EXtis found to be the

expectation of the velocity if a joint or quasi joint distribution of the type in2.1is postulated.

The reason why the time derivative can be taken inside the expectation is that2.1does not have terms of the typeD₁ⁿ, n≥2.

By using thatvXt, x 1/2iLnψt, x/ψ^∗t, xwe have

E X_tⁿY_t

∂

∂t 1 n1

xⁿ¹ρ_Xt, xdx 1 n1

xⁿ¹

−D1

ρ_Xt, xvXt, x dx

xⁿρXt, xvXt, xdx

xⁿψt, xψ^∗t, xD11 2iLn

ψt, x ψ^∗t, x

dx

1 2

ψ^∗t, xxⁿ−iD1ψt, x ψ^∗t, x−iD1

xⁿψt, x dx

1 2

ψ^∗

xⁿpoppopxⁿ

ψdx, popdef −iD1−i ∂

∂x.

4.7

The well-known rule of symmetrization statesEX_tⁿX_t^m → 1/2

ψ^∗p^m_opxⁿxⁿp^m_opψ dx, the rule of Born-Jordan statesEX_tⁿY_t^m → 1/m1

ψ^∗_m

l0p^m−l_op xⁿp_op^l ψ dx, while the Weyl rule statesEXⁿ_tY_t^m → 1/2ⁿ

ψ^∗nl_m

l0x^n−lp_op^mx^lψ dx 43. Form 1 we have more explicitly

E X_tⁿYt

−→ 1 2

ψ^∗

popxⁿxⁿpop

ψ dx, symmetrization,

E Xⁿ_tY_t

−→ 1 2

ψ^∗

p_opxⁿxⁿp_op

ψ dx, Born-Jordan, E

X_tⁿYt

−→ 1 2ⁿ

ψ^∗

xⁿpopnxⁿ⁻¹popxn×n−1

1×2 xⁿ⁻²popx²· · ·popxⁿ

ψ dx, Weyl.

4.8

Only forn0 andn1 the rules do give the same answer ifm1. Notice that when n0, we achieve the same result as in4.5. In general4.7equals the rule of symmetrization and the rule of Born-Jordan, corresponding to the Margenau-Hill44quasi density function and the Born-Jordan quasi density function.

A fundamental conceptual problem in quantum physics or when using 3.1a and 3.1bis to associate something toEX_tⁿY_t². A solution to this operator problem in quantum mechanics can be used for quantization of the kinetic energy in a curved space. Equation2.1 leading to2.4a–2.4cgives

E X_tⁿY_t²

ρ t, x, y

xⁿy²dx dy

v2t, xρXt, xxⁿdx

v_Xt, x²−Πt, x βt

ρ_Xt, xxⁿdx.

4.9

This equation can further be developed if the constitutive model forΠ^Qt, xgiven in3.3a–

3.3eis used. However, logically we can base a quantum theory on the equation set 3.1a and 3.1b and 3.3a together with the association rules EXtY_t → 1/2∂EX²_t/∂t xvXt, xρXt, xdx,EX_tⁿY_t² →

vXt, x²−Πt, x βtρXt, xxⁿdx. Using3.3awith βt 0, or3.3b, implies45

E X_tⁿY_t²

vXt, x²−Πt, x κt

ρXt, xxⁿdx

v_Xt, x²− 1 4ρ_Xt, x

D₁²ρ_Xt, x−

D1ρXt, x₂ ρ_Xt, x

ρ_Xt, xxⁿdx.

4.10

Using againv_Xt, x D₁1/2iLnψt, x/ψ^∗t, xit follows that E

X_tⁿY_t²

−1 4

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎝

D₁ψt, x ψt, x

2

−2

D₁ψt, x ψt, x

D₁ψt, x^∗ ψt, x^∗

D₁ψt, x^∗ ψt, x^∗

2

×ψt, xψt, x^∗xⁿ

−

ψt, x^∗D1ψt, x ψt, xD1ψt, x^∗₂

ψt, x^∗ψt, x xⁿxⁿD²₁ρ_Xt, x

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎠ dx

xⁿ

D1ψt, x^∗

D1ψt, x

−1

4xⁿD₁²ρXt, x

dx.

4.11

We can develop4.11further since E

Xⁿ_tY_t²

xⁿ

D₁ψt, x^∗

D₁ψt, x

−1

4xⁿD₁²ρ_Xt, x

dx

1 4

4xⁿ

D1ψt, x^∗

D1ψt, x

−xⁿ

ψt, xD²₁ψt, x^∗2

D₁ψt, x^∗

D₁ψt, x

ψt, x^∗D²₁ψt, x dx 1

4 xⁿ

−ψt, x

D²₁ψt, x^∗ 2

D1ψt, x^∗ xⁿ

D1ψt, x

−ψt, x^∗xⁿD²₁ψt, x dx

1 4

−ψt, x^∗D²₁

xⁿψt, x

−2ψt, x^∗D₁

xⁿD₁ψt, x

−ψt, x^∗xⁿD₁²ψt, x dx

1 4

ψ^∗

p²_opxⁿ2p_opxⁿp_opxⁿp²_op ψ dx.

4.12

Thus none of the rules stated after4.7are generally in agreement with4.12. However, for n0 all the rules give the same answer and equal the results in4.3. For the particular case n2,4.12equals the Weyl rule. For the cases we have examined we have found that

E X_tⁿY_t^m

1 2^m

ψ^∗

m l

m l0

p_op^n−lxⁿp^l_op ψ dx,

m1⇒E X_tⁿY_t^m

1 2

ψ^∗

popxⁿxⁿpop

ψ dx,

m2⇒E X_tⁿY_t^m

1 4

ψ^∗

p_op² xⁿ2p_opxⁿp_opxⁿp²_op ψ dx.

4.13

More explicitly we can write form2,

E X_tⁿY_t²

−→1 2

ψ^∗

p²_opxⁿxⁿp²_op

ψ dx, symmetrization, 4.14a

E X_tⁿY_t²

−→1 3

ψ^∗

p²_opxⁿpopxⁿpopxⁿp²_op

ψ dx, Born-Jordan, 4.14b

E X_tⁿY_t²

−→ 1 2ⁿ

ψ^∗

xⁿp²_opnxⁿ⁻¹p²_opxn×n−1

1×2 xⁿ⁻²p²_opx²· · ·p²_opxⁿ

ψ dx, Weyl, 4.14c E

X_tⁿY_t² 1

4

ψ^∗

p²_opxⁿ2p_opxⁿp_opxⁿp²_op

ψ dx 4.13. 4.14d

The rule in4.13forn0,m2 can also be found more directly. A joint distribution of the type in2.1allows, as shown, taking the time derivative inside the expectation. This gives

E Y_t²

1 2

∂²

∂t²E X²_t

−E

X_tfXt

EX˙ tY_t−E

X_tfXt

˙

ρXt, xvXt, xxρXt, xv˙Xt, xx

dx−E

XtfXt

−vXt, xxD1

ρ_Xt, xvXt, x

ρ_Xt, xv˙_Xt, xx

dx−E

X_tfXt

ρXt, xvXt, x²xρXt, xv˙Xt, x vXt, xD1vXt, xdx−E

XtfXt

ρ_Xt, xvXt, x²xρ_Xt, xfx xD₁

ρ_Xt, xΠt, x

dx−E

X_tfXt

ρ_Xt, xvXt, x²xD₁

ρ_Xt, xΠt, x

dx

v_Xt, x²−Πt, x

ρ_Xt, xdx.

4.15

Thus to shortly summarize, based on assumptions2.1and3.3a–3.3ewe conclude 4.13which is a new rule of operator ordering.

5. A Quantum Equation Accounting for Friction

Quantum physics does not in general handle friction, but the formulation of the theory in phase space gives such possibilities. The equation for the Wigner function accounting for friction is given by Gardiner and Zollersee37, page 126orAppendix Cfor further details.

The equation is a special case of2.1, to read ρ˙^W

t, x, y −D1

ρ^W

t, x, y y

D₂

D₁Vx εy ρ^W

t, x, y

^∞

k1

i/2^2k 2k1!D^2k1₂

D^2k1₁ Vx ρ^W

t, x, y .

5.1

We set that the classical force isfx, y −D1Vx−εy. The last term in5.1is negligible for the harmonic oscillator, a linear potential, or a free particle. However, there is still a subtle difference with the classical results for the Liouville equation since the possible initial conditions are restricted. The reason is that the initial distribution, for say momentum, should be given through theθt0, p, which is the Fourier transform ofψt0, p. Thus a givenψt0, q gives a unique distribution for momentum. This also implies VarqVarp≥²/4. Inserting fx, y −D1Vx−εyinto2.4a–2.4c choosing unit such that1gives

˙

ρXt, x D1

ρXt, xvXt, x 0,

˙

v_Xt, x v_Xt, xD1v_Xt, x fx−εv_Xt, x D₁

v²_Xt, x−v₂t, x ρ_Xt, x −D1Vx−εD₁St, x D₁

D²₁ρ_Xt, x^1/2 2ρXt, x^1/2 ,

5.2

where we have assumed, most importantly, the same functional form for VarYt/Xtt, x v₂t, x−v_x²t, xas when without friction. We have thatv_Xt, x D₁St, xand again use the Madelung decomposition. It follows after one integration in space that

−1

2D²₁ψt, x Vxψt, x ε1 2iLn

ψt, x ψ^∗t, x

ψt, x iψt, x.˙ 5.3

This is a Schr ¨odinger equation with a linear friction term. It is easily verified that with correlation5.3becomes

−1

2D²₁ψt, x Vxψt, x ε1 2iLn

ψt, x ψ^∗t, x

ψt, x

−βtD1Ln

λψt, xψt, x^∗

iψt, x˙

5.4

which accounts for phase dependency. Further we achieve without correlation

θ˙ p, t

1 2π^1/2

ψt, xExp˙

−ixp dx

1 2π^1/2

−1

2D²₁ψt, x Vxψt, x St, xψt, x

Exp

−ixp dx

p² 2 θ

p, t

ViD1θ p, t

St, iD1θ p, t

.

5.5

We solve5.3 numerically in the next section. Notice that solutions of partial differential equations yield more variety in the solutions than that obtained with nonpartial differential equations since partial differential equations give solutions with an arbitrary number of constants. For other types of nonlinear Schrødinger equations see Weinberg46and Doebner and Goldin47.

6. Simulations and Comparisons of Different Stochastic Approaches

This section compares the Liouville, Ornstein-Uhlenbeck, and quantum solutions for the harmonic oscillator with and without friction. The initial values are

i

ρ^L_Xt0, x ρ^OU_X t0, x, 6.1

ii

ρ^L_Y t₀, y

ρ^OU_Y t₀, y

, ρ^L t₀, x, y

ρ^OU t₀, x, y

ρ_X^Lt0, xρ^L_Y t₀, y

. 6.2

We setfx, y −x−εy. For the quantum solution we choose the initial values iii

ρ_X^Qt0, x ρ_X^Lt0, x ρ^OU_X t0, x. 6.3

Quantum theory uses noρ^Q_Yt0, y. We choose iv

v^Q_Xt0, x v^L_Xt0, x v^OU_X t0, x. 6.4

The initial marginal probability densities for the Liouville and the Ornstein- Uhlenbeck processes are as an example assumed to be Gaussian distributions, that is,

ρ^L_Xt0, x ρ_X^OUt0, x

2πa²_−1/2

e^{−x−x02/2a2},

ρ^L_Y t₀, y

ρ^OU_Y t₀, y

2πb²_−1/2

e^−y2/2b2,

6.5

which imply v^L_Xt0, x v^OU_X t0, x 0. Notice that many different types of ρ_Y^Lt0, y ρ^OU_Y t0, ylead to the samev^L_Xt0, x v^OU_X t0, x v_X^Qt0, x 0.

The analytical solution is for the Liouville and the Ornstein-Uhlenbeck processes with a constant diffusion term, and without friction, given by9

ρXt0, x e^{−x−x0Cost2/2a2Cos2tb2Sin2tq/2t−q/2SintCost} 2π

a²Cos²tb²Sin²t q/2

t− q/2

SintCost_1/2, 6.6a

ρ_Y t₀, y

e^{−y2/2b2Cos2ta2Sin2tq/2tq/2SintCost} 2π

b²Cos²ta²Sin²t q/2

t q/2

SintCost1/2, 6.6b

vXt, x x−x₀Cost

CostSint

b²−a²

qSint a²Cos²tb²Sin²t

q/2 t−

q/2

SintCost−x0Sint. 6.6c

The Liouville solution is achieved when setting q 0 in 6.6a–6.6c. Observe that the variance when q 0 Liouville is steady when a b. When also x0 0, we achieve additionally a steady-state density distribution.

The analytical solution of the quantum equation without friction is given by using the well-known propagator, to read

ψ^Qt, t0, x _∞

−∞ux, x0, t, t₀ψ^Qt0, x₀dx0,

ux, x0, t, t0 2πiSint−t0^−1/2e^{ix2x02Cost−t0−2ixx0/2 Sint−t0},

ψ^Qt0, x ρ^Q_Xt0, x^1/2e^iSt0,x, D1St0, x v^Q_Xt0, x.

6.7