Liouville Equation and Liouville Theorem

The Liouville equation is a fundamental equation of statistical mechanics. It provides a complete description of the system both at equilibrium and also away from equilibrium.

This equation describes the evolution of phase space distribution function for the conservative Hamiltonian system. In essence it is a continuity equation for the flux.

Consider the phase space of a micro-canonical ensemble of a N-particle system (NVE).

Let ρ ()qpt()NN,, () denote the phase space density. If we consider a volume element dp(N)dq(N), then ρ(q(N), p(N), t) dp(N)dq(N) gives the number of representative points (phase points) in the volume element. Liouville equation is an expression of equation of motion of ρ(q(N), p(N),t).

Let us now consider a volume element in phase space. Since the number of system in the ensemble is conserved, the continuity equation for evolution of phase space density is given as,

∂ρ = −∇.J ∂t

= −∇.ρv

N ⎡ ∂∂⎛⎞•• ⎛⎞⎤ =- ∑ ⎢ ⎜⎟ρρqpiiαα+ ⎜⎟⎥ . (1) i=1 ⎣∂∂qpiiαα⎝⎠ ⎝⎠⎦ α =1,3

Applying chain rule one obtains

⎡ ⎛⎞••⎤ ∂∂∂ρρρNN⎛⎞•• ∂∂qp =−qp + −ρ ⎢ ⎜⎟iiαα + ⎥ (2) ∑∑⎜⎟iiαα⎢ ⎥ ∂∂∂tqpii==11⎝⎠iiαα⎜⎟ ∂∂ qp ii αα αα==1,3⎣⎢ 1,3 ⎝⎠⎦⎥

In a Hamiltonian conservative system, the energy is conserved as a function of time and the time derivatives are evaluated by Hamilton’s equations of motion of classical mechanics. The Hamilton’s equations of motion are given by a set of coupled first order partial differential equations,

• ∂H qiα = (3a) ∂piα

• ∂H piα =− (3b) ∂qiα

where qiα is the α-th component of the position qi of the i-th particle and i = 1,2,…….N and α = 1,2,3.

Using Hamilton’s equation of motion, we get

• • ∂ q ∂2 H ∂ p ∂2 H iα = ; iα =− (4) ∂∂∂qqpiiiα αα ∂piiiα ∂∂pqαα

Therefore, the last term (within the third bracket) in equation(2) is identically zero leaving us with

∂∂∂ρρρ⎛⎞•• =−∑⎜⎟qpiiαα + ∂∂∂tqpi,α ⎝⎠iiαα

∂∂∂∂∂ρρρ⎛⎞HH =−∑⎜⎟ + (5) ∂∂∂∂∂tqppqi,α ⎝⎠iiαα ii αα

This is the classical Liouville Equation. It is a highly non-trivial equation, where the momenta and the coordinates of all the N particles of the system are, in principle, coupled with each other.

We can write classical Liouville Equation symbolically:

∂ρ = {}H, ρ (6) ∂t

where {X,H} is the poissin’s bracket, which, for an arbitrary dynamical variable X, is given by

N ⎛⎞∂∂X HXH ∂∂ {}HX, =−∑⎜⎟ − (7) i=1 ⎝⎠∂∂qpii ∂∂ pq ii

Now we define,

LiH= { , } (8)

i{H, } is called Liouville operator, denoted by L and defined as

N ⎡∂HH∂∂∂⎤ iL =−∑ ⎢ ⎥ (9) i=1 ⎣ ∂pii∂∂∂qqp ii⎦

So, we get

∂ρ = {}H, ρ ∂t

= −iLρ (10)

This has the formal solution,

ρρ(te) = −iLt (0) (11)

Liouville Theorem

Let us now consider the total time derivative of the phase space density ρ. Then

dqpρρ∂ N ⎛⎞∂∂ ρ ∂∂ ρ =+∑⎜⎟ + (12) dt∂ ti=1 ⎝⎠∂∂ qii t ∂∂ p t

dρ = 0 (13) dt

This is the Liouville Theorem.

Thus we provide the Liouville Theorem: In conservative system the distribution function is constant along any trajectory in phase space.

Physical Interpretation

The quantity ρ(q(N), p(N),t) dp(N)dq(N) is the probability that at a time t the physical system is in a microscopic state represented by a phase point lying in the infinitesimal 6N- dimensional phase space element dp(N)dq(N). thus the total number of systems in the ensemble is given by the integral over phase space of the distribution,

∫ ρ ()p()NN,qdpdq () () N () N. A normalizing factor is conventionally included in the phase space measure but here has been omitted. A complete knowledge of the probability density enables to calculate the average value of any function of the coordinates and momenta.

The Liouville Equation is the 6N-dimensional analogue of the equation of continuity of an incompressible fluid. It describes the fact that phase points of the ensemble are neither created nor destroyed.

In the simple case of a non-relativistic particle moving in Euclidean space under a force field F with coordinates X and P, Liouvlle’s theorem can be written:

∂ρ P + ..0∇+∇=ρρF (14) ∂tmxP

In Astrophysics this is called the Vlasov equation, or sometimes the Collisionless

Boltzmann Equation. This is used to describe evolution of a large number of collisionless particles moving in a gravitational potential.

In classical statistical mechanics, the number of particles N is very large, (typically of the order of Avogrado’s number, for a laboratory-scale system). Setting ∂ρ/∂t = 0 gives an equation for the stationary states of the system and can be used to find the density of microstates accessible in a given statistical ensemble. The stationary states equation is satisfied by ρ equal to any function of the Hamiltonian H: In particular it is satisfied by

−H /kTB the Maxwell-Boltzmann distribution ρ ∝e , where T is the temperature and kB is the Boltzmann constant.

Significance of Liouville equation

The Liouville equation is valid for both equilibrium and non-equilibrium systems. It is an integral to the proof of fluctuation theorem from which the second law of thermodynamics can be derived. It is also the key component of the derivation of Green-

Kubo relations for linear transport coefficients such as shear viscosity, thermal conductivity or electrical conductivity.

Quantum Liouville Equation (QLE)

In quantum systems, one defines the state of a system in terms of a density matrix. The quantum Liouville equation is straightforwardly obtained from equation(6) by making ρ an operator and transforming the poisson bracket to a commutator.

∂ρ ==−[]HHH, ρˆˆˆρρ (15) ∂t

The density operator ρˆ can have a matrix representation in a basis set.

Let us consider that at some instant, a system is in a completely described state with wave function ψ. ψ can be expanded in terms of the functions ψn(q),where ψn(q) is the normalized wave functions of the stationary states of the system. q denotes the set of all coordinates of the system and n denotes the set of all quantum numbers distinguishing various stationary states.

ψ = ∑cnnψ (16) n

Mean value of density in this state can be calculated from the coefficients cn by means of the formula

* ρ = ∑∑ccnmρ nm (17) nm

Where ρψρψ= * ˆ dq are the matrix elements of ρ ( ρˆ is the operator). nm∫ n m

ρ = ∑∑ wmnρ nm (18) nm

* Where wccmn= n m . The set of quantities wmn is the density matrix and the corresponding statistical operator is wˆ . In this representation, ρ becomes the trace of the operator product wˆ ρˆ .

ρ ==∑(wtrwˆˆρρˆˆ) ( ) . (19) n

the probability that a system is in the n-th state is equal to the corresponding diagonal elements wnn of the density matrix. In classical statistics the distribution function ρ(p,q,t) gives directly the probability distribution of the various values of the coordinates and momenta of the particles of the system. But in quantum statistics, the quantities wnn gives the probability of finding the system in a particular quantum state with no indication of coordinates and momenta of the particles of the system.