statistical mechanics Flashcards
Definition of Temperature
1/KT =(dlng(E)/de)
where K : boltzmanns constnant converts j to k
E: energy of microsttes
g(E) :density of microstates, g(E) e = no. of microstates between E and dE
PEEP
principle of equal equilibrium probability
For isolated system exploring W microstates with the same energy, each is equally probable
Mean energy of a system equation
mean E = sum (EiPi)
Energy when T is small
when T –> 0 E—> 0 at low T system is frozen in ground state which has energy =0
Energy when T is large
when T–> infinity E–>sum of states/number of states
As high T all states have equal boltzmann factors so all equally likely and E is just 1/no. of staates (sum of states)
Gibbs expression for the entropy of a general system
S= -K sumPi (ln) pi
where
pi is the probability that the system is in microstate i
K is the boltzman constant
Entropy of isolated system
by PEEP
pi=1/w for each of w states
0 fo other states
S= -kw(1/w)ln(1/w) = Klnw
Meaning of entropy (based on Gibbs and Boltzmann expression)
s measures (logorithmically) the no. of microstates being explored (klnW) are, more generally (gibbs) the extent to which many microstates are being explored with significant probability.
S at low temps
by third law of thermodynamics s–>0 most real systems have a unique ground state, into which the system will be ‘frozen’ (boltzmann) at low T, w=1
therefore : S=KlnW=0
pressure of a system is given by…?
the thermodynamic relation
P=-(df/dv)T,N
pressure of ideal gas
PV=NRT
relate pressure of a system to pressure of a ideal gas
P=-(df/dv)T,N =+NKT(1/V) –> PV=NKT
Thermodynamic expression for chemical potential
mu = (df/dN)T,V
chemical potential of ideal gas
mu = KT ln (N/V [2πhbar^2/mkT]^3/2)
State the Pauli exclusion principle which applies to identical fermions
Spin S= integers are bosons, integer +1/2 is a fermion, you can’t put more than 1 fermion in each sinlge particle state
fermi dirac equation
f(epsilon) = 1/ e^(epsilon-mu)/KT + 1
what is fermi dirac
The Fermi-Dirac distribution that gives the mean occupation f(c) of a single-particle state of energy ε, in a system of identical non-interacting fermions at temperature T
micro state
quantum state of the entire system. isolated system has fixed energy so only has access to those microstates that have same energy as one another.
g(E) density of microstates in energy
g(E)dE= no. of microstates in range E–> E+dE if E is fixed to within deltaE, W=gdeltaE but for large systems g approx 10^10^23 where as deltaE approaches 10^-20 -10^-1 J so lnw approx lng as -20 –1 are negligable by comparison with 10^23
a physical system that has g(E)=AE^n density of microstates
A system of n harmonics oscillators has, at high temp (KT»hbarW) on energy nKT (classically equipartition) and hence has g(E) of the form given (at high energies)
partition function
Z = sum j exp (-Ei/KT)
Ei : energy of microstates
boltmann function
Pi = exp(-Ei/KT)/Z
no. of q points per unit vol of Q space (sphere)
(L/π)^3)^-1 = V/π^3.
becausse q points form a cubic lattice with spacing π/L
therefore no. points per unit vol of q-space is
((π/L)^3)^-1 = V/π^3. where V=L^3
energy density in q space, sphere ignoring zero point
no. of modes in +ive octant of sphere radius q thickness dq
vol of +ive octant X no of q puv X no. of q modes X energy per mode
vol of +ive octant
4piq^2/8 dq
no of q per unit vol of q modes
v/pi^3
graph epsilon(omega )omega for different temps
at low T, epsilon(omega) is exponentially small throughout visible range
At high T, epsilon(omega is large at red end of visible range but falls off
what is a boson
has a interger of spin and certain atoms and photons
what is a fermion
integer +1/2 of spin electron and proton
For a system of many non-interacting identical particles, state how a fundamental principle of quantum mechanics, depending on whether the particles are bosons or fermions, affects the possible values of the number of particles in a given single-particle quantum state.
Pauli exclusion principle
only one or zero fermions in each single particle state, no restrictions for bosons so 0,1,2,3…
mu in the gibbs disrtibution
mu: chemical potential ith microstate in which energy is Ei
heat resevoir
large system (large heat capacity) whose temperature changes negligibly when energy on the energy scale ofout system is added/subtracted. allowed to exchange energy with our system
microstate
quantum state of our entire system
pi
is the probability that the system is in a microstate i
