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We incorporate randomness into deterministic theories and compare analytically and numerically some well-known stochastic theories: the Liouville process, the Ornstein-Uhlenbeck process, and a process that is Gaussian and exponentially time correlated (Ornstein-Uhlenbeck noise). Different methods of achieving the marginal densities for correlated and uncorrelated noise are discussed. Analytical results are presented for a deterministic linear friction force and a stochastic force that is uncorrelated or exponentially correlated.

Stochastic theories model systems which develop in time and space in accordance with probabilistic laws.^{1}^{2}^{3} 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. Five methods to account for randomness are

For method 1 allowing randomness in the initial values for a particle and for a deterministic noise commonly implies more realistic models of physical situations than, for example, the Liouville process where a realization of the stochastic process is constructed deterministically allowing randomness only in the initial conditions of the particle. In the Stratonovich [

In physics or engineering applications second-order ordinary differential equations are often used as models. Unfortunately, second-order processes are more difficult to address than first-order processes. The second-order differential equation is first written as the mathematical equivalent set of two first-order equations, and then randomness is incorporated into the first-order equations either by Ito or Stratonovich interpretations by defining two stochastic differential equations for the two random variables

In method 2, randomness is implemented without using differential equations and Monte Carlo simulations, by studying the Liouville equation as such with our without added terms, for example, for a particle. In the Liouville process the probability density is a solution of the so-called Liouville equation^{4}

A system with time-uncorrelated noise is usually just a coarse-grained version of a more fundamental microscopic model with time correlation. It is therefore of interest to study models with correlation. Bag et al. [^{5}

This section provides a first order process which accounts for randomness. Assume that the differential equation

Stochastic or nonstochastic integrals can rarely be solved in analytic form, making numerical algorithms an important tool. Assume that

Conceptually, we can easily generate realizations by applying (

We can write (

In Section

The last line in (

With friction we can also compare the variances, to read

Figure

The variances as a function of time assuming linear friction.

We have so far only analyzed first order systems with additive noise. Consider now the continuous time approach with nonlinear noise, to read

We can mimic Gaussian-correlated noise by Gaussian-uncorrelated noise when using the Stratonovich integral for multiplicative noise also. To achieve this we must according to (

Unfortunately bidimensional first order processes or second-order stochastic processes are more difficult to address than one dimensional first order processes. This is so because the position at a given time depends strongly on the velocity. Removing this dependence is tricky. We construct a stochastic interpretation of the bidimensional equation set with additive noise, to read as the Ito stochastic equation

Consider now the second-order process in (

The equation for the joint distribution is more complicated to derive. We achieve by Taylor expansion with only a random force that

By integrating with respect to the position and velocity, respectively, we achieve the equation for the marginal density of velocity and position, to read

This article studies the construction of stochastic theories from deterministic theories based on ordinary differential equations. We incorporate randomness into deterministic theories and compare analytically and numerically some well-known stochastic theories: the Liouville process, the Ornstein-Uhlenbeck process, and a process that is Gaussian and exponentially correlated (Ornstein-Uhlenbeck noise). Different methods of achieving the marginal densities are discussed and we find an equation for the marginal density of velocity and position for a second-order process driven by an exponentially correlated Gaussian random force. We show that in some situations a noise process with exponential correlation can be mimicked by time-dependent uncorrelated noise. We show that a second-order system driven by an exponential time-correlated noise force can be mimicked by adding time-uncorrelated noise both to the position and to the velocity. For such a situation the traditional concept of force loses its meaning.

Following Heinrichs [

We apply a specific Markov process in continuous time and discrete space. Fixing attention to a cell, we define

We do not in this section use the “

We have the following numerical scheme in the Ito formulation (

Say instead that (

Say that the noise is given by

A third method to introduce randomness is by ordinary differential equations for the statistical moments of the probability distribution. These moments can be constructed ad hoc or found by mathematical manipulation of the partial differential equation for the probability density, or simply from a recurrence relation in time.

A fourth method of introducing randomness is the hydrodynamic method. 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 variable (velocity), 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

The fifth method of introducing randomness is quantization rules or Nelson’s [

The Duffing equation is a nonlinear second-order differential equation exemplifying a dynamical system exhibiting chaotic behavior, that is, in the simplest form without forcing

To demonstrate applying Duffing’s method that the methods in this paper work, we propose three methods. First, replacing

Second, replacing

Time

Correlation time parameter

Initial time

Temperature

Stochastic space variable when Stratonovich is applied

Stochastic space variable when Stratonovich is applied

Stochastic space variable when Ito is applied

Stochastic space variable when Ito is applied

Function of a space variable

Deterministic space variable

Deterministic space variable

Time increment

Stochastic variable,

Arbitrary function

Well-behaved function specified exogenously,

Probability density of

Forward infinitesimal operator of the process

Arbitrary function such that

Positive parameter

Dirac delta function

Arbitrary noise function

Parameter, initial value of

Stochastic variable

White noise

Integration variable

Stochastic variable

Deterministic variable

Conditional point probability

Arbitrary function

Absolute value of

Small increment without denomination

Stochastic variable

Time parameter

Probability density.

The authors thank an anonymous referee of this journal and Dr. Joseph McCauley for useful comments on the manuscript.

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 Taylor and Karlin [

In the narrowest sense, a stochastic process has the Markov property if the probability of having state

By “counting up” the different realizations (tracks) in 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

The Liouville equation is present in most standard text books of statistical physics. See, for instance, Lifshitz and Pitaevskĭ [

It quantum physics, the approach by Nelson’s [