Statistical mechanics Definition and 398 Threads

In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles.
Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties such as temperature, pressure, heat capacity, in terms of microscopic parameters that fluctuate about average values, characterized by probability distributions. This established the field of statistical thermodynamics and statistical physics.
The founding of the field of statistical mechanics is generally credited to Austrian physicist Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates, to Scottish physicist James Clerk Maxwell, who developed models of probability distribution of such states, and to American Josiah Willard Gibbs, who coined the name of the field in 1884.
While classical thermodynamics is primarily concerned with thermodynamic equilibrium, statistical mechanics has been applied in non-equilibrium statistical mechanics to the issues of microscopically modeling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.

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  1. damarkk

    Find thermodynamic generalized force

    Assume you have a microscopic pendulum you can suppose is like quantum harmonic oscillator. If the length of pendulum has variation of ##dl##, calculate the work on the pendulum and thermodynamic generalized force. Find also the variation of mean number of extitations. My Attempt Firstly, I...
  2. LightPhoton

    I Volume of momentum space of an ideal gas

    In *An Introduction to Thermal Physics* by Schroeder, while deriving the multiplicity of an ideal gas makes the following statements (image below): Even in quantum mechanics, the number of allowed wavefunctions is infinite. But the number of independent wavefunctions (in a technical sense...
  3. heyhey281

    Thermodynamics: Two gases in a container

    Ideal gas: If the gases are of different type, I would say the entropy stays the same. The total entropy is in both cases just the sum S = S1 + S2, where S1 is the entropy of the first gas and S2 the entropy of the second gas. If the gases are of the same type, I think the entropy change is also...
  4. D

    Heat capacity from dispersion relation

    For me the part of the problem that is giving me issues is obtaining the density of states, since typically how you would calculate D(E) as D(E) = ##\frac{A}{2 \pi} *k*\frac{dk}{dE}## but this shouldn't work since this assumes angular symmetry in k space which this dispersion relationship...
  5. LightPhoton

    How to turn partition sum into an integral?

    In, *An Introduction to Thermal Physics, page 235*, Schroder wants to evaluate the partition function $$Z_{tot}=\sum_0^\infty (2j+1)e^{-j(j+1)\epsilon/kT}$$ in the limit that $kT\gg\epsilon$, thus he writes $$Z_{tot}\approx\int_0^\infty (2j+1)e^{-j(j+1)\epsilon/kT}\,dj$$ But how is this...
  6. R

    B Parcel theory -- how can there be buoyancy with miscible gases?

    Parcel theory holds that as air is heated, it expands. Its density hence decreases and the hot air "floats" upwards, pushed by the colder, more dense air surrounding it. It is an experimental fact that hot air rises, but the explanation from buoyancy seems suspect. In a gas, all motions are...
  7. Yseult

    I Questions about my Understanding of Thermodynamics and Statistical Mechanics

    Good afternoon all, I have two questions to check my understanding/understand better those questions. Why is heat capacity an important quantity in thermodynamics and statistical mechanics? From my understanding, heat capacity is an extensible property so any change in the system would result...
  8. Aethermimicus

    A The relation between ferromagnets, Phi4 and non-linear sigma model

    I'm struggling to understand the relation between phi4 theory,non-linear sigma model and ferromagnets. I've read this in a paper(Phys.Rev.B14(1976)3110):'It is possible to describe the long-distance behavior of the Heisenberg ferromagnets in two different ways:the phi4 theory which corresponds...
  9. M

    Canonical partition function for two coupled oscillators

    I got the hamiltonians for n1 and n2 as (n1+1/2)hw and (n2+1/2)hw. Since the an ensemble is made by N distinguishable pairs of quantum oscillators, the general canonical partition function for the system is 1/N!((sum(-BH(n1))sum(-BH(n2))), where B is the thermodynamic beta. I got to the step...
  10. C

    A Proof of thermodynamic stability condition

    I am watching Kardar's Statistical Mechanics course in my spare time and I am struggling to understand a mathematical detail in the proof of the thermodynamic stability condition. See Eq. I.62 here. The author considers a homogeneous system at equilibrium with intensive and extensive variables...
  11. Rayan

    Second moment of occupation number for bosons

    I tried to show this equality by explicitly determining what $$ \overline{(\Delta \eta)^2} $$ is, but I got a totally different answer for some reason, here is my attempt to solve it, what did I miss?
  12. cianfa72

    I Statistical ensemble in phase space

    Hi, I've a question about the concept of ensemble is statistical physics. Take a conservative system in a given macrostate (e.g. with a given energy): there will be a number of phase space's microstates compatible with the given macrostate. If I understand it correctly, basically the...
  13. J

    Textbook recommendations for second year physics (UK university)

    TL;DR Summary: Which are the best textbooks for the subjects I mention? Those on my reading list are graduate level texts I'm due to start second-year physics in a UK university in October and wanted to prepare well for it (especially after I got a first in year 1). All of the modules I plan...
  14. N

    Other Which Springer books to buy? (QM, GR and statistical mechanics)

    Hello, Springer books are on sale this week so I wanted to buy some textbooks to support my studies and (eventual) future career. I'm an undergrad (in europe) and my courses next year will be QM, GR and statistical mechanics, so I was looking for books about these topics, but any suggestion on...
  15. P

    How to Calculate the Molar Entropy of H2O(g) at 25°C and 1 bar?

    Hi everyone! It's about the following task: Calculate the molar entropy of H2O(g) at 25°C and 1 bar. θrot = 40.1, 20.9K, 13.4K θvib=5360K, 5160K, 2290K g0,el = 1 Note for translational part: ln(x!) = x lnx - x Can you explain me how to calculate this problem?
  16. Heisenberg2001

    How is the Maxwell-Boltzmann Distribution Derived?

    1. ##\vec{p}=m\vec{v}## ##H=\frac{\vec{p}^2}{2m}+V=\frac{1}{2}m\vec{v}^2## ##z=\frac{1}{(2\pi \hbar)^3}\int d^3\vec{q}d^3\vec{p}e^{-\beta H(\vec{p},\vec{q})}## ##z=\frac{Vm^3}{(2\pi \hbar)^3}\int d^3 \vec{v}e^{-\beta \frac{mv^2}{2}}## ##z=\frac{Vm^3}{(2\pi \frac{h}{2\pi})^3}\int d^3...
  17. Ebi Rogha

    Gas temperature in a constant volume

    An insulated container (constant volume, adiabatic) contains an Ideal gas with pressure P1 and temperature T1. We open the container's hatch for a few seconds and let some particles escape from the container, then we close the hatch again. We know container's pressure has reduced by exiting...
  18. rogdal

    Deriving the kinetic energy flux in an effusion process

    I could not find any derivations in the litterature, except for the expected value of the energy flux expression itself: $$\overline{\Phi_{effusion,\epsilon}} = \overline{\dot{N_{ef}}}\overline{\epsilon_{ef}}=\frac{3Nl}{2A}\sqrt{\frac{(k_BT)^3}{2\pi m}}$$ I've started off by calculating the...
  19. rogdal

    Gas in a box with Maxwell-Boltzmann distribution

    I have considered two scenarios: 1) A particle that has just collided with the wall at ##z=L## is moving with a velocity ##v_z<0## moving away from the wall. Hence, the probability that this particle has of colliding again is ##0##, so its distribution is also ##0##. 2) A particle moving with...
  20. A

    How much statistical mechanics is enough for a physicist?

    How much statistical mechanics do I need to know to study QFT, astrophysics, black hole thermodynamics, and other advanced topics? And where should I study it in your opinion? So far I have only read Tong's notes however I don't think it is enough. Some quantum statistical mechanics is also...
  21. C

    I Resources to learn about particles on a grid/mesh

    Hello. I am looking to learn about averaging out a particle gas or any other type of organization of particles within a system or volume that can be approximated onto a grid or mesh where the particles are at a constant distance from each other: https://en.wikipedia.org/wiki/Particle_mesh. I...
  22. G

    I How Is the Partition Function of BaTiO3 Calculated in a Cubic Lattice?

    I have a cubic lattice, and I am trying to find the partition function and the expected value of the dipole moment. I represent the dipole moment as a unit vector pointing to one the 8 corners of the system. I know nothing about the average dipole moment , but I do know that the mean-field...
  23. Bill Dreiss

    How Does Kinetic Theory Connect to Statistical Thermodynamics?

    BSEE MIT.
  24. LCSphysicist

    Statistical mechanics and problem with integrals

    So we have a system of N non interacting particles, on a d-dimensional space, the system is in contact with a bath of temperature T. The hamiltonian is $$H = \sum_{l = 1}^{N} (A_{l}|p_{l}|^{s}+B_{l}|q_{l}|^{s})$$. What is the avarage energy? Now, i have some problems with statistical...
  25. N

    A Derivation of Statistical Mechanics

    Moderator's note: Spin-off from previous thread due to topic change. Because it doesn't work. Bohmian time evolution doesn't involve the coarse graining steps that are used in his calculation. A delta distribution remains a delta distribution at all times and does not decay into ##|\Psi|^2##.
  26. C

    Entropy of spin-1/2 Paramagnetic gas

    As we know, dipole can be only arranged either parallel or anti-parallel with respect to applied magnetic field ## \vec{H} ## if we are to use quantum mechanical description, then parallel magnetic dipoles will have energy ## \mu H ## and anti-parallel magnetic dipoles have energy ## -\mu H##...
  27. K

    I Probability in statistical mechanics

    Suppose we've an isolated box having ##N## classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical. Its said that the probability of having the configuration of ##n## particles in the left side is given as...
  28. D

    I Steam flow rate in 2-chamber steam engine system

    Our system of interest has a duct on the left and a piston chamber on the right that make the shape of the letter T rotated 90º clockwise. The smaller tube on the left is abbreviated as P1 has an unspecified length while the piston chamber is P2. The air in P2 heats up and expands while the...
  29. raisins

    I Phase space integral in noninteracting dipole system

    Hi all, Consider a system of ##N## noninteracting, identical electric point dipoles (dipole moment ##\vec{\mu}##) subjected to an external field ##\vec{E}=E\hat{z}##. The Lagrangian for this system is...
  30. C

    A Effects of a spatially nonuniform diffusion parameter

    Einstein famously derived his relation between the diffusion constant of Brownian motion, particle mobility in a disippative medium, and temperature by considering Brownian motion in a harmonic oscillator potential. The result, $D = \mu k_BT$, is derived assuming that the mobility $\mu$ is...
  31. T

    I What are the biggest problems in the study of complex systems?

    What are the practical purposes of studying complex systems found in nature? And applying statistical methods to them etc.
  32. AndreasC

    I Exceptions to the postulate of equal a priori probabilities?

    Not sure if this is the appropriate forum for this, hopefully if it isn't someone can move it to a more appropriate place. The fundamental postulate of equal a priori probabilities in statistical physics asserts that all accessible microstates states in an ensemble happen with equal...
  33. S

    A All-atom simulation and coarse-grained simulations

    I am currently reading this [paper by Noid et. al.](https://doi.org/10.1063/1.2938860) on the rigorous bridge between atomistic and coarse-grained simulations. In the paper, he defined a linear map from the atomistic coordinates and momenta $$\mathbf{r}^n, \mathbf{p}^n$$ to the coarse-grained...
  34. P

    Understanding basic statistical mechanics formulas

    Firstly, I would like to check my understanding of the first formula: Using velocity distribution = f(v), speed distribution = fs(v): fs(v) = f(vx)f(vy)f(vz)dxdydz, since dxdydz = 4pi*v^2*dv, fs(v) = 4piv^2f(v) The second formula is the confusing one: What does it mean? What is the...
  35. M

    Energy landscape in the two state model (Boltzmann distribution)

    From an excel file I can get the probability of each energy state Εi and I saw at Wikipedia that the probability of each energy is proportional with e^−Εi/KT, from this I find the energy of every micro state. Also from the formula which I found on a paper I can get a curve like the curve...
  36. K

    Probability density in statistical Mechanics

    First of all, I've calculated the partition function:Z=1h3∫e−βH(q,p)d3pd3q=1h3∫e−β(p22m−12mrω2)d3prdrdθdz=2πL(2mπh2β)3/2e12βmω2R2−1ω2mβThe probability of being of one particle in radius $r_0$ is: p(r=r0)=1Z∫e−βHd3pd3q=∫1Z2πL(2mπh2β)3/2eβmrω22rdr So I've thought that because, by definition, the...
  37. A

    Does a statistical mechanics of classical fields exist?

    The usual presentation of classical statistical mechanics are based on the Liouville equation and phase space distribution. This, in turn, is based on the Hamiltonian mechanics of a system of point particles. Real undulatory systems, specially non-linear ones, have to be complex to study...
  38. qnach

    A Partition function in number theory and statistical mechanics

    Is the partition function used in number theory and statistical mechanics the same thing?
  39. tanaygupta2000

    What is the correct formula for the density of states in 2D for normal atoms?

    For getting the density of states formula for photons, we simply multiply the density of states for atoms by 2 (due to two spins of photons). I am getting the 2D density of states formula as :- g(p)dp = 2πApdp/h^2 I think this is the formula for normal particles, and so for photons I need to...
  40. L

    Dispersion relation statistical mechanics

    I know that this is the general case of the Einstein and Debye Model.
  41. phun_physics

    I Derivation of Average Square Energy Fluctuation in a Canonical System

    The canonical ( Boltzmann) distribution law for a canonical system is described the probability of state ##v## by ##P_v = Q^{-1} e^{-\beta E_v} ## where ##Q^{-1}## is the normalization constant of ##\sum_v P_v = 1## and therefore ##Q = \sum_{v}e^{-\beta E_v}##. Chandler then derives ##...
  42. tanaygupta2000

    Statistical Mechanics: Two systems reaching an equilibrium temperature

    First I found partition functions of both the systems and hence total energies of them using above formulas. Z(A) = (1 - e-ε/kT)-1 and Z(B) = (1 + e-ε/kT) Then I equated these values to the given values of total energies. I got: For System A, T(A) = ε/kln(2) > 0 For System B, T(B) =...
  43. tanaygupta2000

    Statistical Mechanics Occupation number

    Upto now I've only dealt with the problems regarding non - degenerate energy states. Since bosons do not follow Pauli's Exclusion Principle, three bosons can be filled in two energy states (say E1 and E2) as: E1 E2 1 boson 2 bosons 2 bosons 1 boson 3 bosons 0 bosons 0 bosons 3...
  44. tanaygupta2000

    Statistical Mechanics: Four non-interacting particles are confined in a box

    Regarding the first part, I proceeded as: nx ny nz 4 0 0 => E1 = 16C 0 4 0 => E2 = 16C 0 0 4 => E3 = 16C 3 1 0 => E4 = 10C 3 0 1 => E5 = 10C 0 3 1 => E6 = 10C 1 3 0 => E7 = 10C 0 1 3 => E8 = 10C 1...
  45. S

    Taylor expansion of an Ising-like Hamiltonian

    For the case when ##B=0## I get: $$Z = \sum_{n_i = 0,1} e^{-\beta H(\{n_i\})} = \sum_{n_i = 0,1} e^{-\beta A \sum_i^N n_i} =\prod_i^N \sum_{n_i = 0,1} e^{-\beta A n_i} = [1+e^{-\beta A}]^N$$ For non-zero ##B## to first order the best I can get is: $$Z = \sum_{n_i = 0,1}...
  46. Saptarshi Sarkar

    Meaning of thermodynamic probability

    I was studying statistical mechanics when I came to know about the Boltzmann's entropy relation, ##S = k_B\ln Ω##. The book mentions ##Ω## as the 'thermodynamic probability'. But, even after reading, I can't understand what it means. I know that in a set of ##Ω_0## different accessible states...
  47. CptXray

    Ideal gas in a cylindrical container

    It looks more like a computational obstacle, but here we go. Plugging all of these to the partition function: $$Q = \frac{1}{N! h^{3N}} \int -\exp(\frac{1}{2m}(p^2_{r}+p^2_{\phi}/r^2+p^2_{z})+gz)d\Gamma=$$ $$= \frac{1}{N! h^{3N}} \int \exp{(\frac{-1}{2m}p^2_{r})}dp_{r_{1}}...dp_{r_{N}}...
  48. S

    What Is a Covariance Matrix in Linear Algebra?

    First, i'd like to apologize for the vague title. Unfortunately my understanding of the question is equally vague. I think the dXd matrix is meant to be a covariance matrix, so the above equation would be some complex constant multiplied by the covariance matrix. The Tr would referring to the...
  49. Jamister

    Adiabatic process in statistical mechanics

    Why is a thermally isolated process that occurs sufficiently slow is necessarily adiabatic and not just reversible process ? Here I mean that the definition of adiabatic process is no change in the entropy of the subsystem, and a reversible process is define by no change of the total entropy of...
  50. L

    A Non-equilibrium Statistical Mechanics of Liquids

    Molecular Transport equations for Liquids are harder to compute than that for gases, because intermolecular interactions are far more important in liquids. A System of equations for particle Distribution function and the correlation functions (BBGKY-Hierarchy) is used in General. For gases, it...
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