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. DarMM

    Criticisms of Jaynes' approach to Statistical Mechanics

    In his well known paper “Information Theory and Statistical Mechanics” Jaynes attempted to formulated statistical mechanics as "nothing more" than the inference theory of many body mechanical systems. I am looking for critiques of this approach. Also of use would be summaries or reviews of the...
  2. 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...
  3. 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?
  4. 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...
  5. A

    Modern uses of classical statistical mechanics?

    Most of the cases when I see applications of statistical mechanics is when Fermi-Dirac or Bose-Einstein statistic are used in condensed matter or the equilibrium equation of neutron stars. Besides the Poisson-Boltzmann equation, I would like to know what are the modern...
  6. I

    I What Determines the Maximum Number of Microstates at Equilibrium?

    ## \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...
  7. 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...
  8. Clara Chung

    Work check and advice on a statistical mechanics problem

    b) Consider P_j(n) as a macrostate of the system, Bosons: P_1(1) = P_2(1) = 1/2*1/2=1/4 ,P_1(2)=P_2(2)=1/2*1/2=1/4 Fermions: P_1(1)=P_2(1)=1 (Pauli exclusion principle), P_1(2)=P_2(2)=0 Different species: P_1(1)=P_2(1) = 2*1/2*1/2=1/2 (because there are two microstates with corresponding to...
  9. Clara Chung

    Statistical mechanics problem of a book shelf

    Equations that might be helpful: Attempt: a) (N_max)!/(n!*(N_max-n)!) i.e. N_max C n b) Total Z = sum n=0 to N_max [(N_max C n) e^(buN)] = (1+e^(bu))^N_max Individual Z = 1+e^(bu*1) = (1+e^(bu)) so individual Z^N_max = total Z c) Now, I use Z to represent the total Z, By equation...
  10. Clara Chung

    Statistical mechanics problem about a paramagnet

    I don't know how to solve part c and d. Attempt: c) B_eff=B+e<M> Substitute T_c into the equation in part b, => (B_eff-B)/e = Nμ_B tanh(B_eff/(N*e*μ_B)) Then? Thank you.
  11. S

    Schools Who are the Top Non-Equilibrium Research Groups in North America and Europe?

    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.
  12. 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...
  13. 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...
  14. 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...
  15. D

    Studying John Baez's list of books math prerequisites?

    my current skills in math are differential eq and linear algebra... and I am 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...
  16. E

    I Derivation of the Onsager symmetry

    Derivation of the Onsager symmetry in many textbooks 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 =...
  17. 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...
  18. T

    Confused on statistical mechanics problem

    Homework Statement A dilute gas of N non-interacting atoms of mass m is contained in a volume V and in equilibrium with the surroundings at a temperature T. Each atom has two (active) intrinsic states of energies ε = 0 and ∆, respectively. Find the total partition function of the gas.Homework...
  19. 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...
  20. Decimal

    Statistical Mechanics: Cooling to Bose-Einstein condensate

    Hello, I have a question regarding the derivation for Bose Einstein condensation. I understand that in a boson gas for high temperatures the expectation value of the total number of particles should equal something like: $$ \langle N \rangle \sim T * \eta(z)$$ With ## z = exp(\frac {\mu} {k_b...
  21. binbagsss

    Statistical Mechanics- moments/cumulants, log expansion

    Homework Statement Using log taylor expansion to express cumulants in terms of moments I have worked through the expansion- ##log(1+\epsilon)= ...## see thumbnail- and that's ok; my question is why does the expansion hold, i.e. all i can see is it must be that ##k## is small- how is this...
  22. Faizan Samad

    Statistical Mechanics And Thermodynamics Textbook.

    This is A very general question. I will be taking physics 112 at Cal (in the future) which is basically stat mech. Almost all professors use Kittel and Kroemer but I’ve heard it’s god awful (I can attest to this having read a little myself). Does anyone know of a secondary textbook that is of...
  23. 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...
  24. ANewPope23

    Studying Learning the non-physics part of Statistical Mechanics

    Hello, this is my first question on PhysicsForum. I am primarily interested in statistics/machine learning. I have recently discovered that many of the ideas used in machine learning came from statistical physics/ statistical mechanics. I am just wondering if it's a bad idea to attempt to learn...
  25. SJay16

    Why Take Quantum and Statistical Mechanics Together?

    At my school, you have to take Quantum mechanics at the same time as Statistical mechanics (co-requisites) in either junior or Senior year as a physics major; why is this? What is the relationship between the 2?
  26. Monsterboy

    In the context of statistical mechanics can anyone define temperature?

    I was told that defining temperature as the "average kinetic energy of the particles in a system" is not accurate enough.
  27. S

    Polymer Chain in Statistical Mechanics

    Homework Statement A polymer chain consist of a large number N>>1 segments of length d each. The temperature of the system is T. The segments can freely rotate relative to each other. A force f is applied at the ends of the chain. Find the mean distance ##\textbf{r}## between the ends...
  28. 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...
  29. 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...
  30. 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...
  31. 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...
  32. binbagsss

    Statistical Mechanics: Can one assume an idealized gas is non-relativistic

    In general when one talks about an idealized gas, should/could one assume it is non-relativistic? (s.t E=p^2/2m will hold) many thanks
  33. 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...
  34. J

    Statistical mechanics and the weather

    Hi all. Where can I find some good introductory sources teaching the use of statistical mechanics to study things like tornado formation or climate in general? I took P. Chem., a while ago now but I'm reviewing the material independently. We used one of Moore's texts, 80's - ish, and in it he...
  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

    What is the true meaning of chemical activity in solutions?

    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. binbagsss

    Taking classical limit question (statistical mechanics )

    1. Homework Statement Question attached. I am looking at the second line limit ##\beta (h/2\pi) \omega << 1 ## 2. Homework Equations above 3. The Attempt at a Solution Q1)In general in an expansion we neglect terms when we expand about some the variable taking small values of the...
  40. 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.
  41. 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...
  42. F

    Was the Ultraviolet Catastrophe a Real Problem or Just a Fake?

    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...
  43. Elizabeth Chick

    Homework Question about Statistical Mechanics

    Homework Statement Consider the system of two large, identical Einstein solids, each with oscillators, in thermal contact with each other. Suppose the total energy of the system is 2 units of the energy quanta, i.e., =2ℏ, (i) how many MACRO-states (e.g., one macro-state corresponds to one...
  44. 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...
  45. 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.
  46. F

    Validity of Two-Fermion System Wavefunction with Quantum Numbers a and b

    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) $$\psi(1,2) = \phi_a(1) \phi_a(2) - \phi_b(1) \phi_b(2) + \phi_a(1) \phi_b(2) - \phi_b(1) \phi_a(2) $$...
  47. 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!
  48. 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.
  49. S

    A Fermi and Bose gas in statistical mechanics

    In statistical mehcanics(pathria, 3rd edition), I have some questions for ideal fermi and bose gases. The textbook handles the approximation for z(=e^βµ) and nλ^3 (n=N/V, λ : thermal de Broglie wavelength). It considers the cases that z<<1, z~1, nλ^3~1,<<1,→0 and so on. In here, I am confused...
  50. 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...
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