What is Statisical mechanics: Definition and 28 Discussions

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  1. D

    Z, <U>, and C for Hagedorn Spectrum

    So to get the partition function I do the integral ##\int \alpha E^{3} e^{(B_{0}-B)E} dE##, which substituting in ##/Delta B = B_{0} - B## is ##Z = \frac{ \alpha E^{3} e^{\Delta B E}}{\Delta B} - \frac{3 \alpha E^{2} e^{\Delta B}}{\Delta B^{2}} + \frac{6 \alpha E e^{\Delta B E}}{\Delta B ^{3}} -...
  2. Juanda

    I Balloon experiment - Classical Physics vs. Statistical Physics

    While reading a similar and deservedly closed post a contradiction came to my mind. The supposed contradiction is related to Statistical Physics where my understanding is only conceptual so correct me where I might be wrong. I remember reading that lightweight gasses can escape Earth's...
  3. P

    A system of independent particles (energy levels)

    Hi guys, Can you give me some feedback on whether my calculation is correct? I applied the formula below (Boltzmann Distribution) but I didn‘t know what to use for the variable z. I don‘t even know if I used the correct equation. Can you help me further? The task is: Consider a system of...
  4. H

    Partition function for a spin i

    ##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...
  5. D

    A Energy hypersurface in a phase space (statistical physics)

    what is the reason for that the energy hypersurfaces in a phase space, which belong to systems with constant energy are closed? ( see picture )
  6. T

    A Brownian Motion (Langevin equation) correlation function

    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 -...
  7. Dom Tesilbirth

    How to find the partition function of the 1D Ising model?

    Attempt at a solution: \begin{aligned}Z=\sum ^{N}_{r=0}C\left( N,r\right) e^{-\beta \left[ -NJ+2rJ\right] }\\ \Rightarrow Z=e^{\beta NJ}\sum ^{N}_{r=0}C\left( N,r\right) e^{-2\beta rJ}\end{aligned} Let ##e^{-2\beta J}=x##. Then ##e^{-2\beta rJ}=x^{r}##. \begin{aligned}\therefore Z=e^{\beta...
  8. anaisabel

    Grand partition function (Volume divided into N spaces)

    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.
  9. T

    Physics Non-equilibrium statistical physics and complex systems

    Is Non-equilibrium statistical physics and complex systems a good area of study to go into? Is it a well respected field? Thank you
  10. Mayan Fung

    I Fermi gas in relativistic limit

    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...
  11. Mayan Fung

    Microcanonical ensemble generalized pressure

    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}...
  12. T

    A special case of the grand canonical ensemble

    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...
  13. P

    Entropy and the Helmholtz Free Energy of a Mass-Piston System

    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...
  14. S

    A When should we use the Langevin equation and when should use Fokker-Planck

    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...
  15. A

    On the width of the kinetic energy distribution of a gas

    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...
  16. Riotto

    I Single-particle phase spaces for a system of interacting particles

    For a system of interacting particles, is it possible to define single-particle phase spaces? If not, why?
  17. QuasarBoy543298

    I Problem with the idea of identical particles in QM

    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...
  18. J

    I Value of beta in Boltzmann Statistics taking degeneracy into account

    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} =...
  19. H

    A What is the functional representation of D(E) for a given energy interval?

    Data = np.array([-1.61032636, -1.23577245, -0.50587484, -0.28348457, -0.18748945, 0.4537447, 1.2338455, 2.13535718]) print("Data is: ", Data) print(Data.shape) n,bins,patches = plt.hist(Data,bins=4) print("n: ",n) print("bins: ",bins) plt.savefig("./DOS")
  20. L

    A Quantum statistical canonical formalism to find ground state at T

    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...
  21. L

    Can we study an odd number sized lattice model at half filling?

    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...
  22. Abhishek11235

    I Changing Summation to Integral

    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?
  23. L

    I What is the relation between chemical potential and the number of particles?

    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...
  24. H

    Probability at a temperature T that a system has a particular energy

    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...
  25. Sizhe

    Finding a Booklist to Learn Nonequilibrium Thermodynamics/Statistical Mechanics

    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...
  26. A

    Scattering dynamics and viscosity

    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...
  27. L

    Looking for a solid Introductory Statisical Mechanics textbook.

    Title says it all really, I'm a second year undergraduate from oxford, and currently the textbook I've been using for stat. mech. is "Concepts in Thermal Physics", which was wirtten by my lecturer. I'd like (ideally) something a bit longer to work through suring the holidays, that would provide...