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EXAMS January 2023 > Advanced Statistical Physics > Flashcards

Advanced Statistical Physics Flashcards

1
Q

What are the axioms of probability theory?

A

The probability of an even happening is real and non-negative.

The probability of something happening is 1.

The probability of one of multiple events happening is equal to the sum of their individual properties (mutually exclusive events).

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2
Q

What is P(AuB)

A

P(A) + P(B) - P(AnB)

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3
Q

What is conditional probability P(A|B)

A

P(AnB) / P(B)

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4
Q

What is Bayes theorem for P(A|B)

A

P(B|A) P(A) / P(B)

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5
Q

What properties do the probabilities of independent events A and B have?

A

P(AnB) = P(A)P(B)
P(A|B) = P(A)

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6
Q

What is the law of total probability of events?

A

P(A) = sum[ P(A|B_i) P(B_i) ]
( mutually exclusive events B_i )
( the union of all events B_i is a complete set)

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7
Q

What is the definition of the cumulative distribution F(x)?
How is the p.d.f. related to the cumulative distribution?

A

The integral of the p.d.f. from -inf to x

The p.d.f. is the derivative w.r.t. x of the cumulative distribution.

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8
Q

What is the expression for variance in terms of moments?

A

2nd moment - first moment^2

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9
Q

What is the characteristic function of a distribution?
How can we recover the p.d.f.?

A

Expectation value of e^(ikx)

p.d.f. = 1/2pi integral[ characteristic function e^(-ikx)]

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10
Q

How can all moments be generated from the characteristic function?

A

Taylor expand e^(ikx) to express in terms of moments.
Recover moments by differentiating w.r.t. k
*there is a (-i)^n term
*evaluated at k=0

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11
Q

What is the binomial theorem?

A

check wall notes

WHERE CAN WE ALSO APPLY THIS?

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12
Q

What is the formula for a geometric series?

A

Check wall notes

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13
Q

How can we calculate the marginal of a multivariate p.d.f.?

A

Integrate w.r.t. the variables we are not interested in.

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14
Q

What property do multivariate p.d.f’s have for independent variables?

A

The p.d.f. is factorisable.

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15
Q

How is the covariance of two variables defined?

What about correlation?

A

<XY> - <X><Y>

cor = cov / sig_x sig_y
</Y></X></XY>

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16
Q

What are the range of values possible for correlation?

A

-1 < < 1

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17
Q

What is the convolution theorem for a variable z that is the sum of variables x_i?

A

The p.d.f. for z is the convolution of p.d.f’s for variables x_i.

The characteristic function for z is the product of characteristic functions for x_i.

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18
Q

What is the law of large numbers?

How can we prove it?

A

For x_i i.i.d. random variables
Z = 1/N sum[x_i] –> mean of x_i distribution

Expand characteristic function for X_i
Use convolution
Use limit form of e
Inverse FT

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19
Q

What is the central limit theorem?

A

For x_i i.i.d. random variables
Z = 1/N sum[x_i]
N –> very large

p.d.f. for Z approaches gaussian with mean mu and variance equal to var(x)/N

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20
Q

What is the form of the gaussian distribution?

A

1/sqrt(2pi var) * e^ - [(x - mu)^2 / 2 var^2]

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21
Q

What are the properties of a stochastic matrix?

A

Elements are transition probabilities q_fi (nonzero)
Columns sum to 1 (preserves normalisation)
Has a unit eigenvalue corresponding to the stationary state. (all other eigenvalues magnitude less than 1)

22
Q

What is the condition for detailed balance?
What is an equivalent property?

A

q_mn P_n = q_nm P_m
Equivalent to reversibility of a markov process

23
Q

Give an example of how Markov chain Monte Carlo works.

A

e.g. ising model average energy is sum over all states hamiltonian * boltzmann distribution.

-Construct markov chain for states generating boltzmann stationary state.
-Apply markov chain and average energy as sum of resulting states hamiltonian (/N)
-Many less terms required in sum.

24
Q

In the example of the Metropolis-Hastings Algorithm, how is a markov chain constructed with a boltzmann stationary state?

A

Using detailed balance condition with boltzmann state distribution probabilities.
-Can ensure this condition by choosing transition probabilities.
min[1, Px’ / Px]

The state x’ is state x with random spin flipped

25
Q

What is the boltzmann distribution?

A

1/Z e^(-beta * energy)

Z is sum of e term over all states

energy is hamiltonian on state for ising model.

26
Q

What is the master equation?

A

Rate of change of P_n =
rate of probability influx to state n
- rate of probability outflow from state n

27
Q

What is the poisson distribution?

A

Check wall notes

28
Q

What property shows diffusion?

A

Variance proportional to time

29
Q

What is the Gillespe Algorithm?
(continuous time Monte-Carlo)

A

Increment by random exp. distributed time.
Choose next state using random number (order states via sum of transition rates)

*check notes

30
Q

What is the general master equation for a non-linear birth-death process?

A

rate of change of <n> =
<W+> - <W-></W-></n>

31
Q

What is the deterministic approximation?

A

Instead of considering <transitionrates>
Consider transition rates(<n>)</n></transitionrates>

-valid only for small fluctuations around <n></n>

32
Q

What is the generating function?

A

F = sum_n [ z^n P_n]

-can taylor expand in z an expression for F obtained via a DE
-compare to sum expression to extract P_n

*simply the first and second derivatives of F w.r.t. z are usually enough.

33
Q

How can we express a conditional probability for a continuous time+space process?

A

P(point | previous path) =
P(whole path) / P(previous path)

34
Q

How can we express the Markovian property for a continuous time+space process?

A

P(point | previous path) =
P(point | previous point only)

35
Q

What is the Lindeberg condition for continuity?

A

Check notes

36
Q

What does the differential Chapman-Kolmogorov equation describe and what terms are involved?
How do we get to the Fokker-Planck equation?

A

Describes time evolution of prob(point | initial point)
-drift term
-diffusion term
-jump term

*assuming no jumps (satisfy Lindeberg condition) gives the Fokker-Planck equation.

37
Q

What is the Fokker-Planck equation?

A

Check wall notes

38
Q

What is the condition on F-P equation solutions given infinite bounds?

A

f(x, t) -> 0 at infinite bounds.

39
Q

What is the form of probability flux for the F-P equation?

A

Consider the continuity equation:
partial_t (f) + partial_x (J) = 0

40
Q

What are absorbing b.c’s for the F-P equation?

A

f = 0 at both bounds

41
Q

What are reflecting b.c’s for the F-P equation?

A

J(a, t) = J(b, t) = 0

42
Q

What are periodic b.c’s for the F-P equation?

A

f same at both bounds
J same at both bounds

43
Q

What is the Weiner process?

A

A Markovian process described by the F-P equation with zero drift and constant diffusion.

44
Q

How can we solve for the Weiner process?

What do we get?

A

Use the characteristic function.

Gives gaussian with mean equal to initial point and diffusive variance.

45
Q

What does the non-differentiability of sample paths refer to for the Weiner process?

A

The Weiner process is continuous but does not have a derivative.
This can be shown by considering the fact that:
<change in W ^2> = delta t

46
Q

What are the general concepts of the Langevin Approach to stochastic processes?

A

Consider path realisations instead of conditional probabilities.
-write a DE including random force term
–>this term has no memory and is considered white noise (differential of the weiner process)

47
Q

What is the issue with the differential Langevin approach?

A

Weiner process derivative does not exist: use an integral equation instead.

–> we now need to define stochastic integration as we have an integral w.r.t. the Weiner process.

48
Q

How can we convert between a Langevin-Ito stochastic DE and the FP equation?

A

Check wall notes

49
Q

What is the formula for Gaussian integral:

inf limits: e^(-ax^2 +bx)

A

Check wall notes

50
Q

How can we approach a polynomial ito stochastic integral?

A

Consider d(W^n) as [W + delW]^n (binomial expansion)

Keep (dW)^2 = dt and dW terms

Rearrange to get integral w.r.t. dW in terms of integrals w.r.t. d(W^n) and dt

51
Q

How can we solve growing network problems?

A

Consider the probability of a new node being connected to a node of degree k.

Then consider the increase/decrease of nodes of degree k at each time step. (separate expression needed for k=1 case)

In the stationary limit, the probabilities do not change over timesteps, use this to solve for the stationary probabilities.

52
Q

What are the key steps in solving fixation/extinction problems?

A

Consider in terms of probabilities that fixation is reached given a state with i individuals.
Construct an equation for relating these probabilities for neighbouring i.

Solve this equation using y as the difference between these neighbouring probabilities.
Also define gamma as ratio of transition rates.
Consider the complete sum of y’s.

EXAMS January 2023 (4 decks)

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