What is Statistical mechanics: Definition and 393 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. M

    Paramagnetic system: computing number of microstates

    Homework Statement We are given a paramagnetic system of N distinguishable particles with 1/2 spin where we use N variables s_k each binary with possible values of ±1 where the total energy of the system is known as: \epsilon(s) = -\mu H \sum_{k=1}^{N} s_k where \mu is the magnetic moment...
  2. J

    Classical Books for Statistical Mechanics self study?

    Hi all, I consider myself a physics self-studier (although I've taken the introductory physics series and more in college), and I'm looking for an introduction to statistical mechanics. My thermal physics class used Schroeder's "Thermal Physics" text, which touches slightly on stat-mech at the...
  3. R

    I Differential number of particles in Fermi gas model

    I'm practicing for the Physics GRE, and came across a question that has me stumped. "In elementary nuclear physics, we learn about the Fermi gas model of the nucleus. The Fermi energy for normal nuclear density (ρ0) is 38.4 MeV. Suppose that the nucleus is compressed, for example in a heavy ion...
  4. J

    A Quantum versus classical computation of the density of state

    Hi, If I consider for instance N non interacting particles in a box, I can compute the energy spectrum quantum mechanically, and thus the number of (quantum) microstates corresponding to a total energy between $E_0$ and $E_0 + \delta E$. In the limit of large quantum numbers, the result is well...
  5. grandpa2390

    Classical Best Textbook for thermodynamics and statistical mechanics

    ok so I am in this class... but my professor is not very helpful. I'm not really caring for the assigned textbook, I want to try a different one. One of the issues is that my professor just threw the textbook away and is doing his own thing, so it is difficult to try and study from the textbook...
  6. F

    B What is Quantum Statistical Mechanics?

    Is Quantum Statistical Mechanics being the application of Quantum Mechanics on the separate particles of bulk matter or the application of QM on whole agregate matter?
  7. kini.Amith

    Classical A source for solved problems in Kinetic theory of gases

    I am doing a course on statistical mechanics and we are using the textbook by Mehran Kardar. We are currently dealing with a chapter on kinetic theory. The problem is, I find the exercise problems to be quite tough (and many of my classmates agree with me), and there are no good solved examples...
  8. jin94

    Statistical Mechanics, partition function for mixing

    Hi! The following image is taken from my note in Stat Mech. Please excuse my ugly handwriting... I copied this from my professor's note on a whiteboard, and I'm not so sure if it is correct. The equations for Z1 (partition function before mixing) and Z2 (partition function after mixing) seems...
  9. J co

    General Extensivity of Entropy

    Homework Statement Which of the following are not extensive functions: S[1] = (N/V)[S[0]+[C][v] ln(T) + R ln(V)] S = (N)[S[0]+[C][v] ln(T) + R ln(V/N)] S[3] = ([N])[S[0]+[C][v] ln(T) + R ln(V/N)] 2. Homework Equations I'm not really sure how to approach this problem. The definition that I...
  10. A

    Virial equation second coefficient derivation

    I've been stuck over this integral for around an hour while studying the derivation of the second coefficient of the virial equation: ∫∫dx1d x2 γ(x1,x2) where γ(x1,x2) is 1 when x1 - x2 < constant. = V∫ dx2 γ(x1,x2) where V is the integral of dx1. Given: periodic boundary condition: x1 + V = x1...
  11. It's me

    Relationship between density and probability in diffusion

    Homework Statement Consider the diffusion of a drop of ink in a water vase. The density of the ink is ## \rho (\vec{r}, t) ##, and the probability ##P(\vec{r}, t)## obeys the diffusion equation. What is the relationship between ##\rho (\vec{r}, t)## and ##P(\vec{r}, t)##? Homework...
  12. B

    Calculating microstates and entropy

    Homework Statement Two identical brass bars in a chamber with perfect (thermally insulating) vacuum are at respective temperature T hot>T cold. They are brought in contact together so that they touch and make perfect diathermal contact and equilibrate towards a common temperature. We want to...
  13. Kiarash

    Fermi's Golden Rule: Countable Quantum States & H-Theorem

    Consider a system with countable quantum states. One can define Jij as the rate of transition of probability from i-th to j-th quantum state. In H-theorem, if one assumes both $$ H:=\Sigma_{i} p_{i}log(p_{i})$$ $$J_{ij}=J_{ji}$$ then they can prove the H always decrease. The latter is Fermi's...
  14. Kiarash

    Why is the logarithm of the number of all possible states of

    Temperature of a system is defined as $$\left( \frac{\partial \ln(\Omega)}{ \partial E} \right)_{N, X_i} = \frac{1}{kT}$$Where Ω is the number of all accessible states (ways) for the system. Ω can only take discrete values. What does this mean from a mathematical perspective? Many people say we...
  15. A

    Statistical Mechanics in GR: Basics & Applications

    Background: I'm just a guy who took some (very old fashioned) undergrad GR course some years ago. I'm only know about the basic stuff and nothing of the more advanced stuff. Question: Is there an statistical mechanics for GR that resembles the one in classical mechanics?, I mean with...
  16. F

    Statistical mechanics - N distinguishable particles

    Homework Statement "A model system consists of N identical ”boxes” (e.g. quantum wells, atoms), each box with only two quantum levels, energies E0 = 0 and E1 = ε What is the number of microstates corresponding to the macrostate with total energy Mε?" The Attempt at a Solution I've done...
  17. akk

    A Normalization constant of Fermi Dirac distribution function

    Fermi-Dirac distribution function is given by f(E)=(1)/(Aexp{E/k_{B}T}+1) here A is the normalization constant? How we can get A? E is the energy, k_{B} is the Boltzmann constant and T is the temperature. thank you
  18. akk

    How to normalize the Fermi-Dirac distribution function?

    n=\int^{\infinity}_{infinity}f(E)d^{3}E i am using this formula,,,but i could not normalize it?Any idea or hint
  19. J

    Classical position/velocity probability

    Probability to find a particle in some region of space is inversely proportional to velocity particle has in that region of space. Let's say we have two cases: one particle has velocity given by v(t)=v0*Cos(w*t), and other by v(t)=v0-v1*Cos(w*t), (v0>v1). Since particle spends more time in...
  20. CassiopeiaA

    Probability Distribution in Ensembles: Explained

    I am confused about the basic idea of probability distribution in ensembles. Given macroscopic properties of the system, a system can have large number of micro states. But isn't the probability of finding a system in any of the micro state is equal? What is then the interpretation of this...
  21. M

    Internal Energy/Helmholtz Free Energy Proof

    Homework Statement Show that (∂(βA)/∂β)N,V = E, where A = E - TS is the Helmholtz Free Energy and E is the Internal Energy. Homework Equations A = E - TS dE = TdS - pdV + ΣUidni β = 1 / (kBT) The Attempt at a Solution (∂(βA)/∂β)N,V = (∂/∂β) * (βE - βTS) (∂(βA)/∂β)N,V = (∂/∂β) * (βE -...
  22. ddd123

    Microcanonical partition function - Dirac delta of operators

    Homework Statement Why is it that the microcanonical partition function is ##W = Tr\{\delta(E - \hat{H})\}##? As in, for example, Mattis page 62? Moreover, what's the meaning of taking the Dirac delta of an operator like ##\hat{H}##? Homework Equations The density of states at fixed energy is...
  23. P

    Suggestion of a topic for a Statistical Mechanics project

    Hi friends, I have to do a semester project (analytic, computational, or both) for my second course in Statistical Physics - a graduate level course with great emphasis on phase transitions. It will be graded just 15% of the final grade (so, it is not necessary to elaborate exhaustively) and it...
  24. C

    Negative amount of particles in statistical mechanics

    Suppose that you have N = \left(\frac{\partial U}{\partial \mu}\right)_{S,V} < 0, supposedly the number of particles, even though the actual number of particles is greater than zero. This means that you can have, in a system subjected to a grand canonical ensemble, less than 0 particle for...
  25. S

    Probability density function of simple Mass-Spring system.

    Homework Statement We know that after long run of simple mass-spring system, there should be a probability of finding the mass at certain points between -A and A.. Obviously in probability of finding the particle near A or -A is higher than finding the particle at 0, because the speed is the...
  26. S

    Thermal Equilibrium and Longitudinal Relaxation

    Homework Statement Problem 6.2 from Magnetic Resonance Imaging: Physical Principles and Sequence Design. Show that M_o \approx \rho_o \frac{s(s+1)\gamma^2\hbar^2}{3kT}B_o Homework Equations M_o = \rho_o \gamma \hbar \frac{\sum m_s e^{m_s(\hbar w_o / kT)}}{\sum e^{m_s(\hbar w_o / kT)}} ...
  27. Jimster41

    Wave mechanics vs Statistical Mechanics

    Is it accurate to say that interference cannot happen in Statistical Mechanics? I know it is considered a wave mechanics phenomenon but aren't waves just highly statistical ensembles, like anything else? I always thought that Fourier says periodic spectra could be summed to create any signal...
  28. Christian Grey

    Can this be explained using statistical mechanics only?

    I want to find photon energy distribution, let's say in a closed cube box,there are 100 photons(suppose 1 photon contains 1 Joule),and there's a mirror on left side and then there's a glass in between and the right side is completely black. I want to find photon energy distribution(like how much...
  29. Coffee_

    Intro statistical mechanics question

    Consider the quote ''The macrostate which corresponds to the highest number of microstates which result in that macrostate, is the state which will be observed.'' Can someone specify in which context this is correct because I'm quite confused by it. If I have an isolated box with N particles in...
  30. I

    Master Thesis Topic: GR & Statistical Mechanics

    Hello! I am a master student, and I am about to start working on my master thesis, which, in my counrty, is a substantial work of 6 months which usually involves original research. I will be supervised by two professors of Statistical Mechanics, who have many research interest. In these days...
  31. ShayanJ

    Statistical mechanics in engineering

    There are some applications of thermodynamics in engineering which I don't bother to talk about. What I want to ask is, are there any problems in engineering for which thermodynamics is not enough and people have to use statistical mechanics? Thanks
  32. J

    Magnetic moment of paramagnetic crystal

    Hello, I've been having some trouble with a paramagnetism problem from my Statistical Mechanics class textbook (F. Mandl, Statistical Physics, 2nd edition, p. 25). The problem is as follows 1. Homework Statement 2. Homework Equations 1. The temperature parameter \displaystyle{ \beta =...
  33. terra

    Volume elements of phase space

    First, two definitions: let ## \varrho (M)## be the probability density of macro states ##M ## (which correspond to a subgroup of the phase space) and ## \mathrm{d} \Gamma ## be the volume element of a phase space. In my lecture notes, the derivation for continuity equation of probability...
  34. alpha358

    Statistical mechanics centers in Germany

    Hi, does anyone know of good places to study masters degree in physics with the main focus on statistical physics ?
  35. muscaria

    The difference between heat and work?

    I realize this question has arisen before in the following thread: https://www.physicsforums.com/threads/difference-between-heat-and-work.461711/ but I felt there may be more room for discussion. I feel that the nature of the effect of heat on physical systems is a rather deep one. If the flow...
  36. T

    Exercises in Statistical Mechanics

    I'm enjoying Lennard Susskind's lectures on Statistical mechanics. I've briefly studied this material before, but I've learned a lot about more practical calculations from these lectures: he's used the partition function to derive all sorts of fun expressions, and to analyse the ideal gas and a...
  37. J

    Maxwell-Boltzmann Spherical Cap

    Homework Statement An ideal gas satisfying the Maxwell-Boltzmann distribution is leaking from a container of the volume V through a circular hole of area A'. The gas is kept in the container under pressure P and temperature T. The initial number density (concentration) is given by n0=N/V...
  38. P

    Prerequisites for graduate statistical mechanics

    Hey guys, I've been looking up on the web and couldn't find prerequisites for graduate level statistical mechanics. What would be the mathematics/physics background needed for the course? Would I need undergraduate statistical physics first? As always, thanks in advance!
  39. C

    The Fundamental Assumption of Statistical Mechanics

    Fundamental assumtption: "a closed system is equally likely to be in any of its g accessible micro- states, and all accessible micro- states are assumed to be equally probable." There's just a few things I don't understand about this, 1. Isn't saying that a closed system is equally likely to...
  40. C

    Statistical Mechanics Mean Field Model

    I don't think I've fully grasped the underlying ideas of this class, so at the moment I'm just sort of flailing for equations to plug stuff into... Homework Statement Show that in the mean field model, M is proportional to H1/3 at T=Tc and that at H=0, M is proportional to (Tc - T)1/2...
  41. C

    Statistical mechanics: Total number of photons in a cavity

    Homework Statement Show that the number of photons in equlibrium at tempertaure t in a cavity of volume V is, N=[2.404 V (t/ħc^3]/Pi^2 The total number of photons is the sum of the average number of photons over all modes n->∑<s> Homework Equations n=Sqrt[nx^2+ny^2+nz^2] ωn=(n Pi c)/L...
  42. Robsta

    Knudson Effusion between two gasses

    Homework Statement Consider two chambers of equal volume separated by an insulating wall and containing an ideal gas, maintained at temperatures T1 = 225K and T2 = 400K. Initially the two chambers are connected by a long tube whose diameter is much larger than the mean free path in either...
  43. Cruz Martinez

    Classical Principles of Statistical Mechanics by Tolman

    I have access to this book at my uni's library. I'm considering to learn from it as my main source. Is it severely outdated? How good is it for learning statistical mechanics?
  44. diegzumillo

    Entropy as log of omega (phase space volume)

    Homework Statement I've seen this problem appear in more than one textbook almost without any changes. It goes like this: Assume the entropy ##S## depends on the volume ##\bar{\Omega}## inside the energy shell: ##S(\bar{\Omega})=f(\bar{\Omega})##. Show that from the additivity of ##S## and the...
  45. diegzumillo

    Probability density for related variables

    Homework Statement Say I calculated a probability density of a system containing m spins up (N is the total number of particles). The probabilities of being up and down are equal so this is easy to calculate. Let's call it ##\omega_m##. Then we define magnetization as ##M=2m-N## and it asks me...
  46. Demystifier

    On the relation between quantum and statistical mechanics

    It is well known that quantum mechanics in the path-integral form is formally very similar to equilibrium statistical mechanics formulated in terms of a partition function. In a relatively recent, very readable and straightforward paper http://lanl.arxiv.org/abs/1311.0813 John Baez (a well known...
  47. L

    Statistical Mechanics of Phase Transitions

    Hello everyone! :) I am studying for an honours course in Statistical Physics. I have stumbled upon the book "Statistical Mechanics of Phase Transitions" by J.M. Yeomans. The layout of the book, as well as information, is exactly what I need. I want to work through the problems and was...
  48. C

    Research Topics in Statistical Mechanics

    It seems like two main applications of statistical mechanics in academia research include the application to solid mechanics/materials and the application to heat transfer. How do statistical mechanics and thermodynamics differ in these two aspects? Is it basically the same theory applied to...
  49. C

    Solving Antiferromagnetic Ising Model on Square Lattice

    Hello, I am trying to work out a mean field theory for an antiferromagnetic Ising model on a square lattice. The Hamiltonian is: ## H = + J \sum_{<i,j>} s_{i} s_{j} - B \sum_{i} s_{i} ## ## J > 0 ## I'm running into issues trying to use ## <s_{i}> = m ## together with the self-consistency...
  50. M

    Density matrix in the canonical ensemble

    Homework Statement We have a quantum rotor in two dimensions with a Hamiltonian given by \hat{H}=-\dfrac{\hbar^2}{2I}\dfrac{d^2}{d\theta^2} . Write an expression for the density matrix \rho_ {\theta' \theta}=\langle \theta' | \hat{\rho} | \theta \rangle Homework Equations...
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