PX000 Foundations of Physics • Reading Task II • David Carvalho

Irreversibility in Classical Mechanics and theArrow of Time

David Carvalho University of Warwick

David.Carvalho@warwick.ac.uk

October 22, 2012

Abstract

In this small report, a simple description of the problem of irreversibility in Physics is givenand an example based upon a scientific paper is used to demonstrate that an arrow of timecan emerge from the formalism, without ad hoc assumptions - such as explicit splitting of timedirections. This result will rely heavily upon the generalisation of the Hamiltonian formalismin Classical Mechanics, by introducing generalised forces are not (necessarily) the result ofpotential differences. Alternative methods are given to prove reversibility in Classical Mechanicsand irreversibility in the functional formalism of Classical Mechanics, where trajectories are notexact and arise from a (normalisable) distribution function ρ. [?]

I. Defining Irreversibility

The key observation when dealing with thetime progression is that natural processes

seem to behave (macroscopically) irreversibly,even though Newton’s equations of motion andmany foundational classical laws are invariantunder time reversal. Furthermore, the well-established empirical and heuristic Second Lawof Thermodynamics offer compelling evidence toconceptualise time as having a direction follow-ing the increase of a statistical quantity in anysystem. The property of Time-Reversal Invari-ance (TRI) is defined as follows: a dynamical lawL is TRI if it is invariant under a time-reversaloperator T such that t 7→ −t. A consequenceof this definition is that if u(t) is a solution toL then so is Tu(t). Furthermore, consider theattractor of a dynamical evolution u(t) to bethe subset of the phase space for which t→ ±∞.Then a solution u(t) of L is reversible if it hasno attractors (see section III)

II. Reversibility withinClassical Mechanics

For a solution x(t) which satisfies Newton’s2nd Law, x(t) = F (x(t)), with initial con-ditions x(0) = x0 and x(0) = v0. Definex(t) := Ψ(t;x0, v0). Let T > 0. Revers-ing the motion of the particle at t = T and

its velocity in a solution y(t) to Newton’s 2nd

Law with y(t) = F (y(t)) and initial conditionsy(0) = x(T ) and y(0) = −x(T ) implies thaty(t) = Ψ(T − t;x0, v0) gives the solution toy(t) = F (y(t)). It is possible to reverse trajecto-ries as they are classically defined by two exact

parameters at a time t, ( ~r(t), ~v(t)).

III. Irreversibility withinFunctional Mechanics

The aim of Functional Mechanics is to modelthe dynamical behaviour of a system by hav-ing a normalisable probability distribution ρ =ρ(q, p, t) (with q, p generalised coordinates andmomenta). The dynamical evolution of such dis-tribution is governed by the Liouville equation:∂ρ∂t +

∑ni=1( ∂ρ∂qi qi + ∂ρ

∂pipi) = 0. This equation

is invariant under T: if ρ(q, p, t) is a solution,so is ρ(q,−p,−t). The fact that q, p are notexactly defined induces an average value of func-tions f(t) defined on the phase-space given by

¯f(t) =∫

Σf(q, p)ρ(q, p, t)dqdp. However, there

is an increase in the dispersion, related to delo-calisation of the system, as the reversal changesρ. This fact supports and accounts for irre-versibility.

One can also find a property which relates towhether a system is irreversible. For ρt = ρt(x)on a phase space Γ, the state is mixing if

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PX000 Foundations of Physics • Reading Task II • David Carvalho

limt→∞ ρt(x) = k, a constant. Liouville’s The-orem is normally used over a smooth manifoldM . One can then define the Liouville measuredµ := dpdq, which is invariant under a phaseflow φt. Then a dynamical system (Γ, φt, dµ) hasthe mixing property if limt→∞ < f, g(φt(x)) >=∫fdµ

∫gdµ, ∀f, g ∈ L2(Γ)

IV. Irreversibility of ahard-discs system

A simple dynamical system is studied and shownto be irreversible. Consider a system of harddiscs frictionless collisions, with diameter andmass 1. Based upon laws of conservation ofenergy and momentum, the equation of motionfor hard discs in matrix form is given by

Vk = Ψkjδ(ψkj(t))∆kj

where• Vk = Vx + iVy is the complex velocity of

the k-th disc along X and Y .• ψkj = (1 − |lkj |)|∆kj |−1, with ∆kj =Vk − Vj being the relative discs’ velocities.

• Ψkj = i(lkj∆kj)(|lkj ||∆kj |)−1

• δ(ψkj) the Dirac delta function.

• lkj(t) = z0kj +

∫ t0

∆kjdt the distances be-

tween the centres of the discs, with z0kj =

z0k − z0

j and z0i the initial coordinates of

the i-th disc.Since the collision forces depend on ∆kj , thesecan’t arise as the gradient of a (potential) scalarfield, forcing Hamilton’s or Liouville’s equationsto be of no use. Using a subsystems-in- equilib-rium argument, a generalised version is obtained:

ddtfm = −fm ∂

∂~pk~Fm where fm = fm(~rk, ~pk, t) is

a (normalised) distribution function for discs in

a m-subsystem, ~Fm =∑Lk=1

∑N−Ls=1 Fmks is the

generalised force acting on the m-subsystem,with k = 1, 2, ..., L running along discs of the m-subsystem and s = 1, 2, .., N − L running alongexternal discs.

Given∑Rm=1

~Fm = 0, the system La-grangian LR follows d

dt (∂LR

∂~rk) − ∂

∂~rkLR = 0

(Euler-Lagrange) and ∂fR∂t + ~rk

∂fR∂~rk

+ ~pk∂fR∂~pk

=

0 (Liouville). As the system is conservative,∑Rm=1∇· ~Jm = 0, with ~Jm = (~rk, ~pk) is the gen-

eralised current vector of a m-subsystem. Sinceddt (

∑Rm=1 log(fm)) = d

dt (log(∏Rm=1 fm)) =

ddt (

∏R

m=1fm)

(∏R

m=1fm)

= 0. This means∏Rm=1 fm = k, a

constant. Consider an equilibrium state; force-fully

∏Rm=1 fm = fR and since

∑Rm=1 F

m = 0

is always verified, one deduces that∏Rm=1 fm =

fR, where fR is the distribution function forthe entire system. The generalised LiouvilleEquation is only in accordance with the Liovilleequation for two situations only:

•∫ t

0( ∂∂~pk

~Fm)→ k for t→ ±∞. This corre-sponds to a irreversibily condition.• ∂

∂~pk~Fm is periodic in time. This corre-

sponds to reversible dynamics.

This fact thus suggests that irreversibilityarises by redistributing volumes between m-subsystems in a phase-space, given that thesystem volume remains invariant. Furthermore,the generalised Liouville equation implies thatnon-potentiality of generalised forces is a neces-sary condition for irreversible dynamics.

IV.References

[1] V.M. Somsikov Irreversibility in Classical Mechanics arXiv:physics/0601038

[2] M.Castagnino, M.Gadella, O.LombardiTime-Reversal, Irreversibility and Arrow of Time in Quantum Mechanics

[3] I.V. Volovich Time Irreversibility Problem and Functional Formulation of Classical MechanicsarXiv:0907.2445

[4] R.Davidson, J.Rae 1970 J. Phys. A: Gen. Phys. 3 128 On the nature of irreversibility in solubleclassical systems

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