# Statisical mechanics Definition and 24 Discussions

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1. ### 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...
2. ### 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 )

17. ### A How to calculate density of states (DOS) from 8 energy eigenvalues of a Quantum model calculated by exact diagonalization?

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")
18. ### 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...
19. ### 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...
20. ### 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?
21. ### 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...
22. ### 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...
23. ### Book(s) to fill the gap from intro thermo to 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...
24. ### 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...