##Z = \sum_{-i}^{i} = e^{-E_n \beta}##
##Z = \sum_{0}^j e^{nh\beta} + \sum_{0}^j e^{-nh\beta}##
Those sums are 2 finites geometric series
##Z = \frac{1- e^{h\beta(i+1)}}{1-e^{h\beta}} + \frac{1-e^{-h\beta(i+1)}}{1-e^{-h\beta}}##
I don't think this is ring since from that I can't get 2 sinh...
So the Langevin equation of Brownian motion is a stochastic differential equation defined as
$$m {d \textbf{v} \over{dt} } = - \lambda \textbf{v} + \eta(t)$$
where the noise function eta has correlation function $$\langle \eta_i(t) \eta_j(t') \rangle=2 \lambda k_B T \delta_{ij} \delta(t -...
equation i need to proof. the N in here, is the avarege number of particles, N0 is the total number of particles,V is total volume, v0 I am not quite sure what it is because it isn't mentioned in the homework, but I am assuming it is the volume of which space.
In a statistical mechanics book, I learned about the degenerate pressure of a Fermi gas under the non-relativistic regime. By studying the low-temperature limit (T=0), we got degenerate pressure is ##\propto n^{5/3}## (n is the density).
And then I was told that in astrophysical objects, the...
In the discussion of the pressure in macrocanonical ensemble, I found in textbook that:
##dW = \bar p dV## (##dW## is in fact d_bar W, yet I can't type the bar)
The derivation goes like:
##\bar p = \frac{1}{Z} \sum_{r} e^{-\beta E_r} (-\frac{\partial E_r}{\partial V}) = ... = \frac{1}{\beta}...
In addition to the homework statement and considering only the case where ##U= constant## and ##N = large## : Can we also consider the definition of chemical potential ##\mu## and temperature ##T## as in equations ##(1)## and ##(2)##, and use them in the grand partition function?
More...
Attempt at a Solution:
Heat Absorbed By The System
By the first law of thermodynamics,
dU = dQ + dW
The system is of fixed volume and therefore mechanically isolated.
dW = 0
Therefore
dQ = dU
The change of energy of the system equals the change of energy of the gas plus the change of energy...
As everyone knows that we can go from Langevin equation to Fokker-Planck equation which gives the evolution of probability density function. But what I don't understand is what is exactly the main difference between them as long as they are both give the variance (which then we can for example...
In these lecture notes about statistical mechanics, page ##10##, we can see the graph below.
It represents the distribution (probability density function) of the kinetic energy ##E## (a random variable) of all the gas particles (i.e., ##E=\sum_{i}^{N} E_{i}##, where ##E_{i}## (also a random...
assume i have a gass made from N identical particles in a box and i want to calculate the probability for k out of N particles to be in the left side of the box.
the problem is ,that if we treat the N particles as identical , each state in which exacly k of the N particles are in the left side...
Hello,
The relationship between entropy ##S##, the total number of particles ##N##, the total energy ##U(β)##, the partition function ##Z(β## and a yet to be defined constant ##β## is:
$$S(\beta)=k_BN \cdot \ln(Z(\beta)) - \beta k_B \cdot U(\beta)$$
Which leads to:
$$\frac{dS}{d\beta} =...
For my own understanding, I am trying to computationally solve a simple spinless fermionic Hamiltonian in Quantum Canonical Ensemble formalism . The Hamiltonian is written in the second quantization as
$$H = \sum_{i=1}^L c_{i+1}^\dagger c_i + h.c.$$
In the canonical formalism, the density...
So one can numerically study (I am interested in exact diagonalization) any 1D lattice model with ##L## sites and ##N## number of particles. At half filling, ##L/N = 2##. My question to a professor was that can we study a system of size ##L = 31## at half filling? He replied yes, there is a way...
This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However,I don't see how. Can anyone justify this change?
Chemical potential is defined as the change in energy due to change in the number of particles in a system. Let we have a system which is defined by the following Hamiltonian:
$$H = -t \sum_i^L c_i^\dagger c_{i+1} + V\sum_i^L n_i n_{i+1} -\mu \sum_i^L n_i$$
where ##c^\dagger (c)## are creation...
Salutations, I'm starting in statistical mechanics and reviewing some related studying cases I would like to understand what occurs in small systems with normal modes of vibration, for example, a small system that has 2 normal modes of vibration, with natural frequencies $$\omega_1$$ and...
Hi, guys
I have posted this question on StackExchange, but no one seems to care answer. Because I don't think this is a simple textbook question, I start my thread here:
I know this is a big question. But as a graduate student, my research is somehow related to nonequilibrium...
I have been studying the statistical mechanics' viewpoint of fluid dynamics by considering the derivation of Navier-Stokes' equations from the Boltzmann equation involving the whole Chapman-Enskog expansion. It is clear that through this process, it is possible to account for the dependence of...