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Randomness in Classical Mechanics and
Quantum Mechanics
Igor V. Volovich
Steklov Mathematical Institute
Gubkin St.8, 119991 Moscow, Russia
email: volovich@mi.ras.ru
Abstract
The Copenhagen interpretation of quantum mechanics assumes the existence of the
classical deterministic Newtonian world. We argue that in fact the Newton determinism
in classical world does not hold and in classical mechanics there is fundamental and
irreducible randomness. The classical Newtonian trajectory does not have a direct
physical meaning since arbitrary real numbers are not observable. There are classical
uncertainty relations: ∆q > 0 and ∆p > 0, i.e. the uncertainty (errors of observation)
in the determination of coordinate and momentum is always positive (non zero).
A “functional” formulation of classical mechanics was suggested. The fundamental
equation of the microscopic dynamics in the functional approach is not the Newton
equation but the Liouville equation for the distribution function of the single particle.
Solutions of the Liouville equation have the property of delocalization which accounts
for irreversibility. The Newton equation in this approach appears as an approximate
equation describing the dynamics of the average values of the position and momenta for
not too long time intervals. Corrections to the Newton trajectories are computed. An
interpretation of quantum mechanics is attempted in which both classical and quantum
mechanics contain fundamental randomness. Instead of an ensemble of events one
introduces an ensemble of observers.
1

1 Introduction
In classical mechanics the motion of a point body is described by the trajectory in the
phase space, i.e. the values of the coordinates and momenta as functions of time, which are
solutions of the equations of Newton or Hamilton, [1].
However, this mathematical model is an idealization of the physical process, rather far
separated from reality. The physical body always has the spatial dimensions, hence a math-
ematical point gives only an approximate description of the physical body. The mathemat-
ical notion of a trajectory does not have direct physical meaning, since it uses arbitrary
real numbers, i.e. infinite decimal expansions, while the observation is only possible, in the
best case, of rational numbers, and even them only with some error. Therefore, we suggest
a “functional” approach to classical mechanics, which is not starting from Newton’s equa-
tion, but with the Liouville equation. This approach can help to explain the infamous time
irreversibility problem, see, for example [2] - [8].
The conventional widely used concept of the microscopic state of the system at some
moment in time as the point in phase space, as well as the notion of trajectory and the mi-
croscopic equations of motion have no direct physical meaning, since arbitrary real numbers
not observable (observable physical quantities are only presented by rational numbers, cf.
the discussion of concepts of space and time in [9] - [14]).
In the functional approach [15] the physical meaning is attached not to a single trajectory
but only to a “beam” of trajectories, or the distribution function on phase space. Individual
trajectories are not observable, they could be considered as “hidden variables”, if one uses
the quantum mechanical notions.
The fundamental equation of the microscopic dynamics of the functional probabilistic
approach is not Newton’s equation, but a Liouville equation for distribution function. It is
well known that the Liouville equation is used in statistical mechanics for description of the
motions of gas. Let us stress that we shall use the Liouville equation even for description of
a single particle in the empty space.
There are many discussions of the time irreversibility problem, i.e. the problem of how to
explain the irreversible behaviour of macroscopic systems from the time-symmetric micro-
scopic laws. The problem has been discussed by Boltzmann, Poincare´, Bogolyubov, Feynman
and many other authors, [2] - [8]. Landau and Lifshiz wrote about the principle of increas-
ing entropy [4]: “Currently it is not clear whether the law of increasing entropy can be in
principle derived from classical mechanics.” Landau speculated that to explain the second
law of thermodynamics one has to use quantum mechanical measurement arguments.
Although the Liouville equation is symmetric in relation to the reversion of time, but
his solutions have the property of delocalization, that, generally speaking, can be interpreted
as a manifestation of irreversibility. It is understood that if at some moment in time the
distribution function describes a particle, localized to a certain extent, then over time the
degree of localization decreases, there is the spreading of distribution function. Delocalization
takes place even for a free particle in infinite space, where there is no ergodicity and mixing.
In a sense, the functional formulation of microscopic dynamics is irreversible in time. Thus
the contradiction between microscopic reversibility and macroscopic irreversibility of the
dynamics disappears, since both microscopic and macroscopic dynamics in the proposed
2

approach are irreversible.
In the functional approach to classical mechanics we do not derive the statistical or
chaotic properties of deterministic dynamics, but we suggest that the Laplace’s determinism
at the fundamental level is absent not only in quantum, but also in classical mechanics.
We show that Newton’s equation in the proposed approach appears as an approximate
equation describing the dynamics of the average values of coordinates and momenta for not
too long time. We calculate corrections to Newton’s equation.
An interpretation of quantum mechanics is attempted in which both classical and quan-
tum mechanics contain fundamental randomness. Instead of an ensemble of events one
introduces an ensemble of observers.
In the next section the fundamentals of the functional formulation of classical and quan-
tum mechanics are presented. Section 3 deals with the free movement of particles and New-
ton’s equation for the average coordinates. Comparison with quantum mechanics is discussed
in Section 4. General comments on the Liouville and Newton equations are given in section
5. Corrections to the Newton equation for a nonlinear system are calculated in Section 6.
The dynamics of the classical and quantum particle in a box and their interrelationships are
summarized in section 7.
2 States and Observables in Functional Mechanics
2.1 Classical mechanics
An exact derivation of the coordinate and momentum can not be done, not only in quantum
mechanics, where there is the Heisenberg uncertainty relation, but also in classical mechanics.
Always there are some errors in setting the coordinates and momenta. There are classical
uncertainty relations: ∆q > 0 and ∆p > 0, i.e. the uncertainty (errors of observation) in the
determination of coordinate and momentum is always positive (non zero). The concept of
arbitrary real numbers, given by the infinite decimal series, is a mathematical idealization,
such numbers can not be measured in the experiment.
Consider the motion of a classical particle along a straight line in the potential field. The
general case of many particles in the 3-dimensional space is discussed below. Let (q, p) be
coordinates on the plane R2 (phase space), t ∈ R is time. The state of a classical particle
at time t will be described by the function ρ = ρ(q, p, t), it is the density of the probability
that the particle at time t has the coordinate q and momentum p.
In [16] it is given a construction of the probability density function starting from the
directly observable quantities, i.e., the results of measurements, which are rational numbers.
2.2 Classical and quantum mechanics
Note that the description of a mechanical system with the help of probability distribution
function ρ = ρ(q, p, t) does not necessarily mean that we are dealing with a set of identically
prepared ensemble of particles. Usually in probability theory one considers an ensemble
of events and a sample space [20]. But we can use the description with the function ρ =
ρ(q, p, t) also for individual bodies, such as planets in astronomy (the phase space in this case
3

the 6-dimensional). In this case one can think on the “ensemble” of different astronomers
which observe the planet. It might be that there is only one “intelligent” observer and an
“ensemble” of different scenario of behaviour of a given object in such a way that one can
deal with an individual quantum phenomenon. From this point of view there is no difference
between “Einstein‘s moon” and “Heisenberg‘s electron”. Actually, it is implicitly always
dealt with the function ρ = ρ(q, p, t) which takes into account the inherent uncertainty in
the coordinates and momentum of the body.
The wave function in quantum mechanics ψ = ψ(q, t) or the density operator actually
depends not only from time t and position q but also from other parameters such as the
form of the potential field and the length of the box as well as from the mass, charge and
the Planck constant. Denote these parameters by ξ. Some of these parameters can be called
the “contextual” variables. We have the wave function ψ = ψ(q, t; ξ). In the functional
formulation of quantum mechanics we introduce a distribution σ(ξ) which describes the
uncertainty in the derivation of these parameters. To get observed quantities we have to
evaluate the average value of ψ or |ψ|2 with σ(ξ). More discussions of functional quantum
mechanics will be presented in a separate work. Note that similar distribution σ(ξ) we have
to introduce already in classical functional mechanics.
The specific type of function ρ depends on the method of preparation of the state of a
classical particle at the initial time and the type of potential field. When ρ = ρ(q, p, t) has
sharp peaks at q = q0 and p = p0, we say that the particle has the approximate values of
coordinate and momentum q0 and p0.
In the functional approach to classical mechanics the concept of precise trajectory of
a particle is absent, the fundamental concept is a distribution function ρ = ρ(q, p, t) and
δ-function as a distribution function is not allowed.
We assume that the continuously differentiable and integrable function ρ = ρ(q, p, t)
satisfies the conditions:
ρ ≥ 0,
∫
R2
ρ(q, p, t)dqdp = 1, t ∈ R . (1)
The motion of particles in the functional approach is not described directly by the New-
ton (Hamilton) equation. Newton’s equation in the functional approach is an approximate
equation for the average coordinates of the particles, and for non-linear dynamics there are
corrections to the Newton equations.
If f = f(q, p) is a function on phase space, the average value of f at time t is given by
the integral
f(t) =
∫
f(q, p)ρ(q, p, t)dqdp . (2)
In a sense we are dealing with a random process ξ(t) with values in the phase space.
2.3 Basic equation for a single particle
Motion of a point body along a straight line in the potential field will be described by the
equation
∂ρ
∂t = −
p
m
∂ρ
∂q +
∂V (q)
∂q
∂ρ
∂p . (3)
4

Here V (q) is the potential field and mass m > 0.
Equation (3) looks like the Liouville equation which is used in statistical physics to
describe a gas of particles but here we use it to describe a single particle.
If the distribution ρ0(q, p) for t = 0 is known, we can consider the Cauchy problem for
the equation (3):
ρ|t=0 = ρ0(q, p) . (4)
Let us discuss the case when the initial distribution has the Gaussian form:
ρ0(q, p) =
1
piabe
−
(q−q0)2
a2 e−
(p−p0)2
b2 . (5)
At sufficiently small values of the parameters a > 0 and b > 0 the particle has coordinate and
momentum close to the q0 and p0. For this distribution the average value of the coordinates
and momentum are:
q =
∫
qρ0(q, p)dqdp = q0 , p =
∫
pρ0(q, p)dqdp = p0 , (6)
and dispersion
∆q2 = (q − q)2 = 12a
2, ∆p2 = (p− p)2 = 12b
2 . (7)
3 Free Motion
Consider first the case of the free motion of the particle when V = 0. In this case the
equation (3) has the form
∂ρ
∂t = −
p
m
∂ρ
∂q (8)
and the solution of the Cauchy problem is
ρ(q, p, t) = ρ0(q −
p
mt, p) . (9)
Using expressions (5), (9),
ρ(q, p, t) = 1piab exp{−
(q − q0 − pmt)2
a2 −
(p− p0)2
b2 } , (10)
we get the time dependent distribution of coordinates:
ρc(q, t) =
∫
ρ(q, p, t)dp = 1√
pi
√
a2 + b2t2m2
exp{−(q − q0 −
p0
m t)2
(a2 + b2t2m2 )
} , (11)
while the distribution of momenta is
ρm(p, t) =
∫
ρ(q, p, t)dq = 1√pibe
−
(p−p0)2
b2 . (12)
5

Thus, for the free particle the distribution of the particle momentum with the passage of
time does not change, and the distribution of the coordinates change. There is, as one says
in quantum mechanics, the spreading of the wave packet. From (11) it follows that the
dispersion ∆q2 increases with time:
∆q2(t) = 12(a
2 + b
2t2
m2 ) . (13)
Even if the particle was arbitrarily well localized (a2 is arbitrarily small) at t = 0, then
at sufficiently large times t the localization of the particle becomes meaningless, there is a
delocalization of the particle.
What role can play the Newton equation in the functional approach? The average coor-
dinate for the free particle in the functional approach satisfies the Newton equation. Indeed,
the average coordinate and momentum for the free particles have the form
q(t) =
∫
qρc(q, t)dq = q0 +
p0
mt , p(t) =
∫
pρm(p, t)dp = p0 . (14)
Note that in the functional mechanics the Newton equation for the average coordinates is
obtained only for the free particle or for quadratic Hamiltonians with a Gaussian initial
distribution function. For a more general case there are corrections to Newton’s equations,
as discussed below.
4 Comparison with Quantum Mechanics
Compare the evolutions of Gaussian distribution functions in functional classical mechanics
and in quantum mechanics for the motion of particles along a straight line. The scene of work
for the functional classical mechanics is L2(R2) (or L1(R2)), and for quantum mechanics -
L2(R1).
The Schrodinger equation for a free quantum particle on a line reads:
i~∂ψ∂t = −
~2
2m
∂2ψ
∂x2 . (15)
Here ψ = ψ(x, t) is the wave function and ~ is the Planck constant. The density of the
distribution function for the Gaussian wave function has the form
ρq(x, t) = |ψ(x, t)|2 =
1
√
pi
√
a2 + ~2t2a2m2
exp{−(x− x0 −
p0
m t)2
(a2 + ~2t2a2m2 )
} . (16)
We find that the distribution functions in functional classical and in quantum mechanics
(11) and (16) coincide, if we set
a2b2 = ~2 . (17)
If the condition (17) is satisfied then the Wigner function W (q, p, t) for ψ corresponds to the
classical distribution function (10) , W (q, p, t) = ρ(q, p, t) .
Gaussian wave functions on the line are coherent or compressed states. The compressed
states on the interval are considered in [17].
6

5 Liouville Equation and the Newton Equation
In the functional classical mechanics the motion of a particle along the stright line is described
by the Liouville equation (3). A more general Liouville equation on the manifold Γ with
coordinates x = (x1, ..., xk) has the form
∂ρ
∂t +
k
∑
i=1
∂
∂xi (ρv
i) = 0 . (18)
Here ρ = ρ(x, t) is the density function and v = v(x) = (v1, ..., vk) - vector field on Γ. The
solution of the Cauchy problem for the equation (18) with initial data
ρ|t=0 = ρ0(x) (19)
might be written in the form
ρ(x, t) = ρ0(ϕ−t(x)) . (20)
Here ϕt(x) is a phase flow along the solutions of the characteristic equation
x˙ = v(x) . (21)
A system in the phase space R2N with coordinates q = (q1, ..., qN ), p = (p1, ..., pN) is
described by the Liouville equation for the function ρ = ρ(q, p, t)
∂ρ
∂t =
∑
i
(∂V (q)∂qi
∂ρ
∂pi
− pimi
∂ρ
∂qi
) . (22)
Here summation goes on i = 1, ..., N . The characteristics equations for (22) are Hamilton’s
equations
q˙i =
∂H
∂pi
, p˙i = −
∂H
∂qi
, (23)
where the Hamiltonian is
H =
∑
i
p2i
2mi
+ V (q) . (24)
Emphasize here again that the Hamilton equations (23) in the current functional approach
to the mechanics do not describe directly the motion of particles, and they are only the
characteristic equations for the Liouville equation (22) which has a physical meaning. The
Liouville equation (22) can be written as
∂ρ
∂t = {H, ρ} , (25)
where the Poisson bracket
{H, ρ} =
∑
i
(∂H∂qi
∂ρ
∂pi
− ∂H∂pi
∂ρ
∂qi
) . (26)
7

6 Corrections to Newton’s Equations
In section 3, it was noted that for the free particle in the functional approach to classical
mechanics the averages coordinates and momenta satisfy the Newton equations. However,
when there is a nonlinear interaction, then in functional approach corrections to the Newton’s
equations appear.
Consider the motion of a particle along the line in the functional mechanics. Average
value f of the function on the phase space f = f(q, p) at time t is given by the integral (2)
f(t) =< f(t) >=
∫
f(q, p)ρ(q, p, t)dqdp . (27)
Here ρ(q, p, t) has the form (20)
ρ(q, p, t) = ρ0(ϕ−t(q, p)) . (28)
By making the replacement of variables, subject to the invariance of the Liouville measure,
we get
< f(t) >=
∫
f(q, p)ρ(q, p, t)dqdp =
∫
f(ϕt(q, p))ρ0(q, p)dqdp . (29)
Let us take
ρ0(q, p) = δ(q − q0)δ(p− p0) , (30)
where
δ(q) =
1√
pie
−q2/2 , (31)
q ∈ R,  > 0.
Let us show that in the limit → 0 we obtain the Newton (Hamilton) equations:
lim
→0
< f(t) >= f(ϕt(q0, p0)) . (32)
Proposition 1. Let the function f(q, p) in the expression (27) be continous and inte-
grable, and ρ0 has the form (30). Then
lim
→0
∫
f(q, p)ρ(q, p, t)dqdp = f(ϕt(q0, p0)) . (33)
Proof. Functions δ(q) form a δ-sequence in D
′(R) [18]. Hence we obtain
lim
→0
∫
f((q, p))ρ(q, p, t)dqdp = lim
→0
∫
f(ϕt(q, p))δ(q − q0)δ(p− p0) = f(ϕt(q0, p0)) , (34)
that was required to prove.
Now calculate the corrections to the solution of the equation of Newton. In functional
mechanics consider the equation, see (3) ,
∂ρ
∂t = −p
∂ρ
∂q + λq
2∂ρ
∂p . (35)
8

Here λ is a small parameter and we set the mass m = 1. The characteristic equations have
the form of the following Hamilton (Newton) equations:
p˙(t) + λq(t)2 = 0 , q˙(t) = p(t) . (36)
Solution of these equations with the initial data q(0) = q, q˙(0) = p for small t has the form
(q(t), p(t)) = ϕt(q, p) = (q + pt−
λ
2 q
2t2 + ..., p− λq2t+ ...) (37)
Use the asymptotic expansion δ(q) in D′(R) for → 0, compare [7, 19]:
δ(q) = δ(q) +
2
4 δ
′′(q) + ... , (38)
then for → 0 we obtain corrections to the Newton dynamics:
< q(t) >=
∫
(q + pt− λ2 q
2t2 + ...)[δ(q − q0) +
2
4 δ
′′(q − q0) + ...] (39)
·[δ(p− p0) +
2
4 δ
′′(p− p0) + ...]dqdp = q0 + p0t−
λ
2 q
2
0t2 −
λ
4 
2t2 .
Denoting the Newton solution
qNewton(t) = q0 + p0t−
λ
2 q
2
0t2 ,
we obtain for small , t and λ:
< q(t) >= qNewton(t)−
λ
4 
2t2 . (40)
Here −λ4 2t2 is the correction to the Newton solution received within the functional
approach to classical mechanics with the initial Gaussian distribution function. If we choose
a different initial distribution we get correction of another form.
We have proved
Proposition 2. In the functional approach to mechanics the first correction at  to the
Newton dynamics for small t and λ for equation (36) has the form (40).
Note that in the functional approach to mechanics instead of the usual Newton equation
m d
2
dt2 q(t) = F (q) , (41)
where F (q) is a force, we obtain
m d
2
dt2 < q(t) >=< F (q)(t) > . (42)
The task of calculating the corrections at  for Newton’s equation for mean values is similar
to the problem of calculating semiclassical corrections in quantum mechanics.
9

7 Dynamics of a Particle in a Box
Dynamics of collisionless continuous medium in a box with reflecting walls is considered by
Poincare and in [8]. This studied asymptotics of solutions of Liouville equation. In functional
approach to mechanics, we interpret the solution of the Liouville equation as described the
dynamics of a single particle. Here we consider this model in the classical and also in the
quantum version for the special case of Gaussian initial data. In particular we obtain that
a single free particle in the box behaves like a gas with the Maxwell type distribution.
7.1 Dynamics of a classical particle in a box
Consider the motion of a free particle on the interval with the reflective ends. Using the
method of reflections [18], the solution of the Liouville equation (8)
∂ρ
∂t = −
p
m
∂ρ
∂q
on the interval 0 ≤ q ≤ 1 with the reflective ends we write as
ρ(q, p, t) =
∞
∑
n=−∞
[ρ0(q −
p
mt+ 2n, p) + ρ0(−q +
p
mt+ 2n,−p)] , (43)
where it is assumed that the function ρ0 has the Gaussian form (5).
One can show that for the distribution for coordinates ρc(q, t) =
∫
ρ(q, p, t)dp one gets
the uniform limiting distribution (pointwise limit): limt→∞ ρc(q, t) = 1 . For the distribution
of the absolute values of momenta (p > 0) ρa(p, t) = ρm(p, t) + ρm(−p, t) , where
ρm(p, t) =
∫ 1
0
ρ(q, p, t)dq ,
as t→ ∞ we get the distribution of the Maxwell type (but not the Maxwell distribution):
lim
t→∞
ρa(p, t) =
1√
pib [e
−
(p−p0)2
b2 + e−
(p+p0)2
b2 ] .
7.2 Dynamics of a quantum particle in a box
The Schrodinger equation for free quantum particle on the interval 0 ≤ x ≤ 1 with reflecting
ends has the form
i~∂φ∂t = −
~2
2m
∂2φ
∂x2 (44)
with the boundary conditions φ(0, t) = 0, φ(1, t) = 0, t ∈ R . Solution of this boundary
problem can be written as follows:
φ(x, t) =
∞
∑
n=−∞
[ψ(x+ 2n, t)− ψ(−x+ 2n, t)] ,
10

where ψ(x, t) is some solution of the Schrodinger equation. To get an observable quantity
we have to compute the average value with the distribution σ(ξ) (see Sect.2). Such average
values will demonstrate an irreversible behaviour for large time. If we choose the func-
tion ψ(x, t) in the form, corresponding to the distribution (16), then one can show that in
the semiclassical limit for the probability density |φ(x, t)|2 the leading term is the classical
distribution ρc(x, t).
8 Conclusions
In this paper the functional formulation of classical mechanics is considered which is based
not on the notion of an individual trajectory of the particle but on the distribution function
on the phase space.
The fundamental equation of the microscopic dynamics in the functional approach is
not the Newton equation but the Liouville equation for the distribution function of a single
particle. Solutions of the Liouville equation have the property of delocalization which ac-
counts for irreversibility. It is shown that the Newton equation in this approach appears as
an approximate equation describing the dynamics of the average values of the positions and
momenta for not too long time intervals. Corrections to the Newton equation are computed.
If we accept the functional approach to classical mechanics then both classical and quan-
tum mechanics contain fundamental randomness. It requires a reconsideration of the usual
interpretation(s) of quantum mechanics. Some remarks on these questions we made in Sect.2.
Interesting problems related with applications of the functional mechanics to statistical me-
chanics, field theory, singularities in cosmology and black holes we hope to consider in further
works.
9 Acknowledgements
I am grateful to many colleagues for valuable discussions. Some results of this paper were
presented at the QTRF5 (Vaxjo, Sweden) and QBIC2009 (Tokyo) conferences. The work
is partially supported by grants NS-3224.2008.1, RFBR 08-01-00727-a, 09-01-12161-ofi-m,
AVTSP 3341, DFG Project 436 RUS 113/951.
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