Statistical mechanics

Microscopic System: A system of atomic dimension or of smaller size is called microscopic system. Eg: a molecule

Macroscopic System: A system which is large enough to be observable in the ordinary shape is called a macroscopic system.

Microstate: If the states of all constituent particles of the system is specified then it is called the microstate of the system. Generally the coordinate (x,y,z) and the momentum (Px,Py,Pz) of each molecules of the system within the limit of the specification.

Macrostate: If the number of molecules or phase points in each cell of the phase space is specified then it is called macrostate of the system. The specification means to the specification of N, V and E where the symbol have their usual meaning.

Phase space: The specification of microstate of the system containing N particles required 3N position co-ordinates i.e q1, q2,q3...P3N. The set of coordinates(qi,pi; i=1,2....3N) may be regarded as a point of 6N dimensions which is called phase space(𝜏 - space)

If H(pi,qi) be the Hamiltonian of the system, then

qi = δ H(pi,qi)/ δpi and pi = -δ H(pi,qi)/δqi

As the time passes the set of co ordinates (qi,pi) undergoes a continuous change. Then the representative points in the phase space traces a trajectory which is called phase space trajectory.

Note: It is to be noted that the two phase points forming trajectories cannot be at a point at a same time.

µ- Space and density function

The phase space of a single molecule is called a µ space i.e representing 3 position and 3 momentum coordinates. The density function ρ(q,p,t) is the probability that the state point (q,p) lies in the volume element at time 't'. Since state point must always lie somewhere in the phase space.

Using normalization condition

∫ ρ (p,q,t)dp dq = 1

𝜏 space

Concept of ensemble:

The physical quantities of interest in a thermodynamic system are usually the time average over the trajectory in phase space. They are very large number of particles of the order 10^23 in general calculation of time average of physical quantities is a very tedious task.

Gibb's along with Boltzmann (the founder of statistical mechanics) made a great advance by suggesting the alternative way of calculating the averages. There could be a large number of phase points in 𝜏 space, all of which correspond to same microscopic state. Each of these points would represent same system and these system will have the feature that their micro structures are different but their bulk (macroscopic) behaviour is same.

This random collection of such systems each of which corresponds to the same microscopic thermodynamics state but has different microstate is called an ensemble. The great importance of the ensemble arises from the remark of Gibb's that average over a period of time for a given system are equivalent to the average over the ensemble at one instant of time.

Types of ensemble:

As regards to the particle applications three types of ensemble are listed:

1. Microcannonical Ensemble

It is the collection of independent system in which the collected system are isolated that is they exchange neither energy nor mass with one another. This implies that the total number of particles N and total energy E remains constant. Thus in microcannonical ensemble , N,V and E are constant.

2. Cannonical Ensemble

Cannonical ensemble corresponds to closed isothermal system which exchange energy but not mass with one another and this implies that the number of particles N, volume V and temperature T remains constant.

3. Grand Cannonical Ensemble

Grand Cannonical ensemble corresponds to open isothermal system which exchange energy and mass with one another. This implies that this collection chemical potential µ, volumeV and temperature T remains constant.

Surface Energy

The locus of all points in phase space satisfying the condition H(p,q)= E defines a surface called the surface energy of E.

Liouville's theorem

It states that the density of group of points remains constant along their trajectories in the phase space and extension of phase space remains constant with time.

Proof: Let us consider an arbitrary volume element in the phase space which is enclosed by surface area dσ. Let ρ (p,q,t) be the density function. Then the rate at which the number of representative points in this volume increase with time is written as

Now one possible way of satisfying eq 9 is to assume that ρ which is already shown that it has no explicit dependencies on time and to satisfy 9, the density function must ne independent of coordinates (q,p) over the region of phase space.

Hence, this eq 9 reflects the principle of conservation of extension in phase space.

Hamiltonian formulation of Mechanic Legendre dual transformation

Lagrangian to Hamilton formulation corresponds to changing the variables in our mechanical function from (qj, q̇j, t) to (qj, pj, t). The procedure for switching variable is provided by Legendre transformation.

Consider a function of only two variables f(x,y). Then,

df = ∂f dx/∂ x + ∂f dy/∂y

= udx + vdy...........1

u= ∂f/∂x, v= ∂f/∂y....2

To change the basic of description x,y to a new distinct variable u,y so that differential quantities are expressed in terms of the differentials du and dy. Let g be a function of u and y define by the equation,

g= f - ux......3

dg = df - udx- xdu

= udx +vdy-udx-xdu

= vdy -xdu.........4

which is exactly as

x= -∂g/∂u, v=∂g/∂y...............5

which are in the effect of equation 2

The Legendre transformation is frequently used in thermodynamics. For example the enthalpy X is a function of entropy S and the pressure p with the properties that

∂X/∂s = T, ∂X/∂p = -V

dX = Tds + Vdp

where T and V are temperature and Volume respectively.

Conservation theorem and Physical significance of Hamilton:

According to definition, a cyclic coordinate qj is one that does not appear explicitly in the Langrangian. If aj does not appear in L then

ṗj = ∂L/∂qj = 0, pj = constant

But pj is constant, then from Hamilton's cannonical equation of motion, we get

ṗj = ∂H/∂qj = 0 which implies qj would not appear in H.

Conversely if a generalized coordinate does not occur in H, the conjugate momentum is conserved. The momentum conservation theorem can thus be transferred to the Hamiltonian formulation with no more than a substitution of H for L.

Physical significance:

Like Lagrangian, H also posses the dimensions of energy but in all circumstances it is not equal to total energy E. Restriction for this equality. i.e. E=H are

1. The system be conservative one. i.e P.E is coordinate dependent and not velocity dependent

2. Coordinate transformation equations be independent of time so that

Σ pjqj = 2T

These conditions are necessary for Hamiltonian H to represent the total energy E. It may happen that though coordinate transformation equation involve time explicity yet L be still independent of time, then

H= Σ pj q̇j - L(qj,q̇j)

will be a constant. That is for such cases H will continue to be a constant of motion but not the total energy.

It can be shown that if it does not occur in Lagrangian L then the Hamiltonian H will also not involve t. Since

H= H(qj,pj,t)

dH/dt = Σ ∂H/∂qj q̇j + Σ ∂H/∂pj ṗj + ∂H/∂T

Also from cannonical equation,

∂H/∂qj = - ṗj and ∂H/∂pj = q̇j

Therefore, dH/dt = - Σ ṗj q̇j + Σ ṗj q̇j+ ∂H/∂t = ∂H/∂t

But ∂H/∂t = - ∂L/∂t

If L does not involve time t, then ∂L/∂t=0, show that ∂H/∂t= 0 which implies dH/dt = 0

This means that t will also not appear in H, i.e t is cyclic Hamiltonian for it to be a constant of motion.