In a paper published in year 2000 Aykut Barka and colleagues estimated the probability of occurance of a major earthquake in Istanbul in the following 30 years as 60%. (The paper may be accessed through this link.) (a) Assuming a constant event rate for the process, determine this rate. (In your answers to this question, use "years" as your basic unit of time.) (b) What is the expected waiting time <t>? (c) Find the fluctuation Δt = √(<t2> - <t>2) in this variable. (d) In how many years does the probability of occurance reach 90%? (e) Assume that the event rate was zero at the time the paper was published, but has been increasing linearly since then, i.e. ω(t) = αt, where t = 0 corresponds to year 2000. Find the value of α that would result in the same 60% event probability in 30 years. [Hint: Solve the equation dP/dt = -ω(t) P(t) to solve for P(t) for a general ω(t).] (f) Repeat parts (b), (c), and (d) above for this rate
Suppose {x} are random numbers distributed uniformly between 0 and π/2.
Now consider the numbers generated by the function y=sin(x). Clearly, 0<y<1.
(a) What is the probability distribution function P(y)=?
(b) Use uniform random numbers generated by computer for x, and obtain the random numbers sin(x).
Compare this with the expected distribution.
[Hint: Since histogram distributions may contain large fluctuations, it is more feasable to compare
the theoretical and computational density functions ∫oy P(y) dy ]
Please submit your computer codes as well.
A particle moves with the dynamics associated with the Langevin equation dx/dt = v dv/dt = -α v3 + η(t) where α is a constant and η(t) represents a Wiener process noise. The properties of η(t) are such that ∫P(η) dη = 1 ∫P(η) η dη = 0 ∫P(η) η2 dη = D (a) Find the corresponding Fokker Planck equation for P(x,v,t) (b) Set dP/dt = 0 and ∂P/∂x = 0 to find the corresponding equilibrium velocity distribution Peq(v).
Find the Wigner function corresponding to the one-dimensinal wavefunction ψ(x) = A exp[ -(x/xo)^2 + i po x/ℏ ] where xo and po are constants with appropriate units. A is the normalization constant.
(a) N non-interacting electrons are placed in a three-dimensional harmonic oscillator potential.
Using the separation of variables in rectangular coordinates, identify the energy states associated
with the single-particle states.
(b) Find the Fermi energy (the maximum energy of the particles in the ground state).
(You can assume that ℏω is very small compared to the system energy so that summations
may be approximated by integrals.)
(c) What is the total energy of the system with N particles?
Find the variation of the magnetization due to a system of N particles, each of which has two electrons contributing to the magnetization. The Hamiltonian for the pair of electrons looks like H = γ (S1 + S2)⋅B - J S1⋅S2 where S1 + S2 are the spin operators for the two electrons, B is the magnetic field, γ and J are the appropriate interaction constants. Hint: Take B to be uniform in the z-direction. Using the total spin eigenstates in the trace operation, obtain Q = Tr exp(-β H) and use βγ[S1z+S2z] = -∂/∂B ln QThis is what I get:
Consider a 2xN classical spin system made up of spins s and t
s1 s2 s3
⋯ - o - o - o - ⋯
| | |
⋯ - o - o - o - ⋯
t1 t2 t3
with the reduced hamiltonian -βH = K ∑i (siti + sisi+1 + titi+1)
Construct the transfer matrix and find its largest eigenvalue to relate it to the partition function.
(You will most likely need computer code to find the eigenvalue numerically.) Use Numerical
differentiation to find the specific heat as a function of K.