Phase space

A hypothetical space Molecule as a point in space Any translational motion of particle have six dimension 3 positional –x, y and z 3 conjugate momenta coordinates Px,Py,Pz Volume element can be dx,dy,dz

Px,py and pz can be plotted in three mutually perpendicular directions –and consider 3D volume dpx,dpy,dpz. Thus a six dimensional space can be imagined- in which dx,dy,dz,dpx,dpy,dpz are element of volume Position of a point in this space described by set of six coordinates x,yz,px,py,pz- called phase space Element of volume in space is called cell

Meaning of point in phase understood based on uncertainty principle For this divide a phase into small six dimensional cells with sides dx, dy, dz, dpx, dpy, dpz Volume of these cells given as dτ = dxdydzdpxdpydpz

By uncertainty principle dxdpx ≥ h, dydpy ≥ h, dzdpz ≥ h So dτ ≥ h3 A point in phase space is considered to be cell whose minimum volume is of the order h3 A particle can be understood as being located in such a cell centered at some location instead of being precisely at a point

Classical mechanics- motion of particle in phase space Quantum mechanics- motion described in quantum state So volume of phase space and volume of quantum state are to be related Statistical thermodynamics- relates volume of phase space and volume of quantum state

Thermodynamic probability For a macro state defined as- number of micro states corresponding to that macro state- (OR) total number of micro states in a macro state- total number of different ways in which the given system in a specific thermodynamic state may be realised In general this is a large number denoted by W or P Example – crystal at 0K

Consider two cells in phase space i and j and four phase points or molecules a,b,c and d If Ni and Nj are number of phase points or molecules in the cell , then possible macro states are Ni 4 3 2 1 0 Nj 0 1 2 3 4

For each of these macro states there will be different micro states Consider micro states corresponding to macro state Ni = 3 and Nj = 1 There will be four microstates Cell i abc abd acd bcd Cell j d c b a

Thermodynamic probability for this macro state is W = 3+1 = 4 In general thermodynamic probabilty W = N!/ n1!n2!....ni! ln W = ln N -∑ln ni Apply Stirling theorm lnW = NlnN –N – Nni ln ni +∑ni = NlnN –N - ∑niln ni +N = NlnN - ∑ni ln ni since N = ∑ni

Condition for maximum probability is δlnW = δ[NlnN - ∑nilnni = 0 = ∑(1+ lnni) δn = 0

Statistical equilibrium An ensemble is said to be in statistical equilibrium if it obeys the following conditions The probabilities of finding phase points in the various regions of phase should be independent of time ii) The average values for the properties of the system in the ensemble should also be independent of time

Mathematically (∂ρ/∂t)q, p = 0 ------(1) q = x, y, z p = px, py, pz So density ρ is to be independent of time at all points in the phase space for an ensemble in statistical equilibrium To evaluate density ρ - it is assumed to be a function of some property ex. Energy- which in turn expressed as a function of q and p ie represented by Hamiltonian function

If this property is α Then ρ = ρ(α) -------(2) as α is a function of the coordinates and momenta with a value for a given system- it will change with time So dα/dt = ∑(∂α/∂qi)qi’ + (∂α/∂pi)pi’ = 0-----(3)

By Liouvilles theorem (∂ρ/∂t)q,p = -∑[(∂ρ(α)/∂qi)qi’+ (∂ρ(α)/∂pi)pi’] – (4) Equ(2) can be written as ∂ρ/∂qi = ∂ρ/∂α x ∂α /∂qi and ∂ρ/∂pi = ∂ρ/∂α x ∂α /∂pi ------ (5) Substitute (5) in (4)

(∂ρ/∂t)q,p = -∑[(∂ρ/∂α x ∂α /∂qi )qi’ + (∂ρ/∂α x ∂α /∂pi )pi’] = - ∂ρ/∂α ∑[(∂α /∂qi )qi’ + (∂α /∂pi )pi’] (6) Substitute (3) in (6) (∂ρ/∂t)q,p = - (∂ρ/∂α)(0) = 0 (7) From this equation - ρ is a function of some property of the ensemble independent of time , ensemble is in statistical equilibrium

Relationship between entropy and probability There is correlation between entropy of a system and extent of order or disorder Similar relation can be between entropy and probability Ex. Two globes containing gases separated by stop cock

Open the stopcock –based on II law of tdics- strong probability is gas distributes uniformly in both globes If only single molecule present- probability that it will be found in either of the globes- so probability is ½ So for N molecules – probability that all molecules present in original globe is (1/2)N. ie probability extremely small

This is also the probability that the molecules will return spontaneously to the globe after having been distributed uniformly throughout the globes Probability of distribution of considerable number of molecules in the available space is very large.

Probability of Uniform distribution of gas in both containers is large At constant temperature- spontaneous process in which gas fills uniformly in the whole available volume is thus associated with large increase in the probability of the system

In general – all spontaneous process represent changes from a less probable to a more probable state. Such process are accompanied by increase of entropy- there is a connection between entropy and probability in a given state

If ‘S’ is entropy and ‘W’ is probability then S can be represented as a function of W ie S = f(W) Consider two systems- SA,SB,WA,WB as entropies and probabilities If the systems are combined- probabilities has to be multiplied and entropies added

SAB = SA + SB = f(WA x WB) Since SA = f(WA) and SB = f(WB) Then f(WA)+f(WB) = f(WA x WB) To satisfy this the function must be logarithmic So S = klnW + constant k is a constant with dimension of entropy- assumed to be zero by Boltzmann and Planck So the equation is called as Boltzmann –Planck equation

Thermodynamic probability is defined as the total number of different ways in which a given system may be realised at specified thermodynamic state In a system of perfect solids at absolute zero all molecules in their lowest energy state are arranged in a definite manner in a crystal At these conditions – only one way to realise the system –thus W is unity- so entropy must be zero by B-P equation- this is in agreement with III law of tdics.

There are solids in which molecules may be arranged in different ways in a crystal at absolute zero Such solids are not perfect crystals in the sense of III law of tdics- and entropies are not zero Similar consideration apply to solid solutions and glasses.