Statistical mechanics Definition and 128 Discussions

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. 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##...
  2. 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...
  3. raisins

    I Phase space integral question

    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...
  4. 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...
  5. 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.
  6. 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...
  7. 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...
  8. 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...
  9. 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...
  10. tanaygupta2000

    Density of States in 2D

    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...
  11. 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 ##...
  12. 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}...
  13. 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...
  14. 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}}...
  15. 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...
  16. J

    A Liouville's theorem and time evolution of ensemble average

    With the Liouville's theorem $$\frac{{d\rho }}{{dt}} = \frac{{\partial \rho }}{{\partial t}} + \sum\limits_{a = 1}^{3N} {(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{d{p_a}}}{{dt}} + \frac{{\partial \rho }}{{\partial {q_a}}}\frac{{d{q_a}}}{{dt}})} = 0$$ when we calculate the time evolution...
  17. A

    Classical Companion book to Huang's Statistical Mechanics

    My professor will be using Huang's Statistical Mechanics next semester and I have been reading a lot of polarizing reviews. Does anyone recommend a book to read parallel to Huang's to better understand the material and that discusses the same topics in similar fashion?
  18. hilbert2

    A Summation formula from statistical mechanics

    I ran into this kind of expression for a sum that appears in the theory of 1-dimensional Ising spin chains ##\displaystyle\sum\limits_{m=0}^{N-1}\frac{2(N-1)!}{(N-m-1)!m!}e^{-J(2m-N+1)/kT} = \frac{2e^{2J/kT-J(1-N)/kT}\left(e^{-2J/kT}(1+e^{2J/kT})\right)^N}{1+e^{2J/kT}}## where the ##k## is the...
  19. I

    I Most probable macrostate

    ## \Omega(E_1)## is the number of microstates accessible to a system when it has an energy ##E_1## and ##\Omega(E_2)## is the number of microstates accessible to the system when it has an energy ##E_2##. I understand that each microstate has equal probability of being occupied, but could...
  20. S

    Average speed of molecules in a Fermi gas

    My first most obvious attempt was to use the relation ##<\epsilon> = \frac{3}{5}\epsilon_F## and the formula for kinetic energy, but this doesn't give the right answer and I'm frankly not sure why that's the case. My other idea was to use the Fermi statistic ##f(\epsilon)## which in this case...
  21. S

    Schools Best non-equilibrium groups?

    I realize the question is quite broad but what research groups working on statistical physics, stochastic processes, and complex systems are generally considered the best? Would like to know about Europe and America alike.
  22. CharlieCW

    One-dimensional polymer (Statistical Physics)

    Homework Statement Consider a polymer formed by connecting N disc-shaped molecules into a onedimensional chain. Each molecule can align either its long axis (of length ##l_1## and energy ##E_1##) or short axis (of length ##l_2## and energy ##E_2##). Suppose that the chain is subject to tension...
  23. F

    Kinetic theory of gases and velocity correlations

    I have been reading up on the kinetic theory of gases, and I'm unsure whether I have correctly understood why particle velocities become correlated after colliding. Is it because during the collision they exchange momentum and thus their velocities (and hence trajectories) are altered in a...
  24. F

    I Is something wrong with my understanding of Liouville's Theorem?

    One version of Liouville’s Theorem for non-dissipative classical systems, governed by a conserved Hamiltonian, is that the volume in phase space (position-momentum space) of an ensemble of such systems (the volume is the Lebesgue measure of the set of points where the ensemble’s density is...
  25. D

    Studying John Baez's list of books math prerequisites?

    my current skills in math are differential eq and linear algebra... and im about to start reading Feynman lectures of physics and planning to read all John Baez's recommended books.. after reading Feynman's, what would be the next best thing to do? learn more math? or jump already to core...
  26. E

    I Derivation of the Onsager symmetry

    Derivation of the Onsager symmetry in many text books and papers is as follows: First, assume that the correlation function of two state variables,##a_i## and ##a_j## satifsies for sufficiently small time interval ##t## that $$\langle a_i(t) a_j(0) \rangle = \langle a_i(-t) a_j(0) \rangle =...
  27. JD_PM

    Computing the second virial coefficient

    Homework Statement ##b_2## is the second virial coefficient Homework Equations Virial expansion: $$P = nkT(1 + b_2 (T)n + b_3 (T)n^2...)$$ $$b_2 = -\frac{1}{2} \int dr f(r) $$ r is the distance vector. $$f(r) = e^{-\beta \phi(r)} - 1$$ The Attempt at a Solution $$b_2(T) = 2 \pi r^2...
  28. M

    I General Concepts About Fermi-Dirac Distribution

    Hello! Thanks for your time reading my questions. When I was studying quantum statistical mechanics, I get so confused about the relations between Pauli's exclusion principle and the Fermi-Dirac distributions. 1. The Pauli's exclusion principle says that: Two fermions can't occupy the same...
  29. M

    I Fermi Energy Calculations About Non Parabolic Dispersions

    Greetings! It is easy to understand that for a free electron, we can easily define the energy state density, and by doing the integration of the State density* Fermi-Dirac distribution we will be able to figure out the chemical potential at zero kelvin, which is the Fermi-Energy. Hence, we can...
  30. D

    Velocity correlations and molecular chaos

    I’ve been reading up about Boltzmann transport equations, and the concept of molecular chaos has come up, in which one assumes the velocities of particles are assumed to be uncorrelated. I’m a bit confused about the concept though. In what sense do the velocities become correlated in the first...
  31. S

    Using Maxwell-Boltzman Statistics

    Homework Statement Determine if the classical approximation (Maxwell-Boltzman statistics) could be employed for the following case: a) Electron gas in a metal at 2.7K (cubic metal lattice of spacing 2Å) Homework Equations Maxwell-Boltzman statistics are acceptable to use if the de broglie...
  32. S

    Change in molar entropy of steam

    Homework Statement Calculate the change in molar entropy of steam heated from 100 to 120 °C at constant volume in units J/K/mol (assume ideal gas behaviour). Homework Equations dS = n Cv ln(T1/T0) T: absolute temperature The Attempt at a Solution 100 C = 373.15 K 120 C = 393.15 K dS = nCvln...
  33. S

    Numerical integration of sharply peaking functions

    Homework Statement ∫ e1000((sinx)/x) dx [0 to 1000 : bound of integration]. Solve this integral of a sharply peaked function without a calculator. Homework Equations I'm doing this in relation to statistical thermodynamics - I think I need to use Sterling's Approximation or a gamma function...
  34. J

    Expectation Value of a Stochastic Quantity

    Homework Statement I'm working on a process similar to geometric brownian motion (a process with multiplicative noise), and I need to calculate the following expectation/mean; \langle y \rangle=\langle e^{\int_{0}^{x}\xi(t)dt}\rangle Where \xi(t) is delta-correlated so that...
  35. zexxa

    Canonical ensemble of a simplified DNA representation

    Question Form the canoncial partition using the following conditions: 2 N-particles long strands can join each other at the i-th particle to form a double helix chain. Otherwise, the i-th particle of each strand can also be left unattached, leaving the chain "open" An "open" link gives the...
  36. R

    Harmonic Oscillator and Volume of Unit Cell in Phase Space

    Long time no see, PhysicsForums. Nevertheless, I have gotten myself into a statistical mechanics class where the prof is pretty brutal and while I can usually manage, this problem finally has me stumped. I'd like to be nudged in the right direction, not outright given the answer if possible. I...
  37. E

    Meaning of Chemical Activity

    Hello - I am wondering abut the meaning of chemical activity. Most definitions are something along the lines of "effective concentration," which is fine until you have a real concentration in a lab, and you don't know if you need to calculate the concentration or "effective" concentration of the...
  38. Ron Burgundypants

    Modeling an Einstein solid that is coupled to a paramagnet

    I'm working on a project at university to calculate the magnetocaloric effect of dysprosium. This will be done using a new technique designed at the university of which its not necessary to go into detail about. In short, the Dy is placed in a solenoid, through which a current runs, the current...
  39. NFuller

    Statistical Mechanics Part II: The Ideal Gas - Comments

    Greg Bernhardt submitted a new PF Insights post Statistical Mechanics Part II: The Ideal Gas Continue reading the Original PF Insights Post.
  40. A

    I Density of States -- alternative derivation

    I am trying to understand the derivation for the DOS, I get stuck when they introduce k-space. Why is it necessary to introduce k-space? Why is the DOS related to k-space? Perhaps if someone could come up to a slightly different derivation (any dimensions will do) that would help. My doubt ELI5...
  41. F

    The Ultraviolet Catastrophe

    Max Planck formulated the quantum hypothesis, that electromagnetic radiation was emitted from heated bodies only in quanta of energy E = hf, where f was the frequency of the radiation and h was a constant now called “Planck's Constant”, in order to solve the Ultraviolet Catastrophe...
  42. B

    Trying to reconcile two definitions of Entropy

    My question is regarding a few descriptions of Entropy. I'm actually unsure if my understanding of each version of entropy is correct, so I'm looking for a two birds in one stone answer of fixing my misunderstanding of each and then hopefully linking them together. 1) A measure of the tendency...
  43. G

    Logarithm and statistical mechanics

    Hello, I'll try to get right to the point. Why and how does logarithmic dependence appear in statistical mechanics? I understand that somehow it is linked with probabilities, but I can not quite understand.
  44. F

    Fermion-system quantum state

    Homework Statement Is the statement ”Given a two-fermion system, and two orbitals φ labeled by quantum numbers a, b, the two-body wavefunction (1,2 represent the particle variables) $$\psi(1,2) = \phi_a(1) \phi_a(2) - \phi_b(1) \phi_b(2) + \phi_a(1) \phi_b(2) - \phi_b(1) \phi_a(2) $$...
  45. F

    I Check invariance under time-reversal?

    Hi! How do I check if the equation of motion of the particle, with a given potential, is invariant under time reversal? For a 2D pointlike particle with potential that is e.g $$V(x) = ae^(-x^2) + b (x^2 + y^2) +cy', where a,b,c >0$$ Can it be done by arguing rather then computing? Thanks!
  46. NFuller

    Statistical Mechanics Part I: Equilibrium Systems - Comments

    Greg Bernhardt submitted a new PF Insights post Statistical Mechanics Part I: Equilibrium Systems Continue reading the Original PF Insights Post.
  47. A

    I What is the formal definition of a Universality Class?

    Hi guys, I have been reading some of the literature recently concerning the Kardar-Parisi-Zhang equation and the words "universality" and "KPZ universality class" keep appearing. I already did a cursory wikipedia search on the subject, but it did not make much sense to me. Can you please...
  48. B

    A Can KE be reformulated using |v| instead of v^2?

    While doing some calculations on v_rms using the Maxwell-Boltzmann distribution, I noticed that v_rms and v_avg are pretty similar (https://casper.berkeley.edu/astrobaki/index.php/File:MaxwellSpeedDist.png). In fact, really it's just the choice of using the 1-norm (|v|_avg) vs. 2-norm sqrt(v^2...
  49. chrononaut 114

    "From your data, is the bandgap of ZnSe direct or indirect?"

    (urgent) Hi, This question was apart of an assignment sheet that I was given in 'Experimental Physics III' after having completed and obtained data for the practical called 'The Bandgap Energy of Semiconductor ZnSe'. Cheers Below is some screenshots of the (Matlab-processed) data we obtained...
  50. Wrichik Basu

    Other Books for Non-Equilibrium Statistical Mechanics

    Can anyone refer to some good book on Non-Equilibrium Statistical Mechanics? The book should contain the basics, and can go up to any advanced level. Any level of math is acceptable.
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