What is Statistical thermodynamics: Definition and 40 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. cianfa72

    I Feynman's Brownian Ratchet analysis

    Hi, as in a previous thread I would like to better understand the Feynman's analysis of brownian ratchet as described here: https://www.feynmanlectures.caltech.edu/I_46.html https://en.wikipedia.org/wiki/Brownian_ratchet Consider the case in which the two boxes (i.e. heat baths) are at the same...
  2. 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...
  3. 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...
  4. Dario56

    I Concept of Thermal Equilibrium in the Context of Canonical Ensemble

    Canonical ensemble can be used to derive probability distribution for the internal energy of the closed system at constant volume ##V## and number of particles ##N## in thermal contact with the reservoir. Also, it is stated that the temperature of both system and reservoir is the same, i.e...
  5. Dario56

    I How Can Internal Energy of the Canonical Ensemble Change (Fluctuate)?

    Canonical ensemble is the statistical ensemble which is applicable for the closed system in contact with the reservoir at constant temperature ##T##. Canonical ensemble is characterized by the three fixed variables; number of particles ##N##, volume ##V## and temperature ##T##. What is said is...
  6. Dario56

    I Derivation of the Canonical Ensemble

    One of the common derivations of the canonical ensemble goes as follows: Assume there is a system of interest in the contact with heat reservoir which together form an isolated system. Heat can be exchanged between the system and reservoir until thermal equilibrium is established and both are...
  7. Dario56

    I Boltzmann Entropy Formula – Derivation

    Boltzmann entropy definition is given by: $$ S = k_B lnW $$ where ##W## is the weight of the configuration which has the maximum number of microstates. This equation is used everywhere in statistical thermodynamics and I saw it in the derivation of Gibbs entropy. However, I can't find the...
  8. S

    I Microstates of an atom in an energy state

    I have a question about a sentence in the book Introduction to Thermal Physics (Daniel v. Schroeder). So in chapter 6, Schroeder talks about an atom isolated. This means its energy is fixed. The atom is in some state. The energy states of the atom have degenerated. All microstates with that...
  9. patric44

    Question in Bose-Einstein statistics

    iam not getting why in bose statistics the number of ways to arrange ni particles in gi degenerate states is = (gi+ni-1) ? and why do we divide by ni factorial , and gi factorial .
  10. Sizhe

    The number of vibrational modes g(v)dv in Debye theory

    Homework Statement Show that, in the Debye theory, the number of excited vibrational modes in the frequency range ##\nu## to ##\nu+d\nu##, at temperature T, is proportional to x2e-x, where ##x=h\nu/kT##. The maximum in this function occurs at a frequency ##\nu'=2kT/h##; hence ##\nu'→0## as...
  11. D

    Deriving Fermi-Dirac Distribution misunderstanding

    Homework Statement The actual question was deriving Bose-Einstein, but I got confused on the F-D example. I'm basically following the method given here. Homework Equations [All taken directly from the above link] Taylor series: The Attempt at a Solution So after that third equation...
  12. H

    Integral constant for internal energy of an ionic liquid

    Integral constant for internal energy of ionic liquid I have a question, and I will be really grateful if someone helps me. I have a polynomial equation for internal energy which I calculated by integration an equation of state formula, which is based on density. But, because I calculated this...
  13. A

    Classical Best Statistical Mechanics books for studying for qualifier?

    Does anyone have any good books, or other references, that they would recommend for studying for the thermodynamics & statistical mechanics portion of graduate qualifying exams? I didn't have any undergrad Stat Mech and my grad prof/class was really not good, to the point that I didn't really...
  14. Toby_phys

    Boltzmann vs Maxwell distribution?

    So I worked through the Boltzmann distribution and got: $$ P\propto e^{\frac{-E}{k_BT}} $$ Where $E$ is the energy. So surely this means the kinetic energy (and therefore speed) of particles is distributed over a Boltzmann distribution. Or in equation: $$ P\propto e^{\frac{-mv^2}{2k_BT}} $$...
  15. PePaPu

    Thermodynamic assembly - Statistical Thermodynamics

    Homework Statement Consider a model thermodynamic assembly in which the allowed (nondegenerate) states have energies 0, ε, 2ε, 3ε.The assembly has four distinguishable (localized) particles and a total energy U = 6ε. Tabulate the nine possible distributions of the four particles among the...
  16. M

    Lattice Models for Fluids - Regular Solution Model

    1. Problem Statement: For the regular solution model, develop the equations for the compositions of the coexisting phases in a binary system and plot the phase boundary as a function of χ/RT.2. This question stems from Sandler's Introduction to Applied Statistical Thermodynamics. The Attempt...
  17. adriplay

    Heat capacity in gas simulation

    I have a simulation with a bunch of particles with volume bouncing around in a box with no interaction between them, a hard-sphere gas. Initially, they all have the same momentum |p|=√(2⋅m⋅2/3⋅k⋅T) to have the average kinetic energy 3/2⋅k⋅T. I'm asked to add a constant energy flux to the system...
  18. D

    Shannon entropy of logic gate output.

    Homework Statement A particular logic gate takes two binary inputs A and B and has two binary outputs A' and B'. I won't reproduce the truth table. Suffice to say every combination of A and B is given. The output is produced by A' = \text{NOT} \ A and B' = \text{NOT} \ B . The input has...
  19. A

    Classical Books for statistical thermodynamics and oscillatory motion

    Can someone recommend me some good textbooks or articles that contain or focus on statistical thermodynamics and/or oscillatory motion (preferably with advanced math, not just stories)?
  20. D

    Mean energy of system with ## E = \alpha |x|^n ##

    Homework Statement If the energy ##E## of a system behaves like ## E = \alpha |x|^n##, where ## n =1, 2, 3, \dots ## and ## \alpha > 0##, show that ## \langle E \rangle = \xi k_B T ##, where ##\xi## is a numerical constant. Homework Equations $$ \langle E \rangle = \frac{ \int_{- \infty}^{...
  21. V

    Statistical Thermodynamics (multiple questions)

    Homework Statement 1.) For N particles in a gravity field, the Hamiltonian has a contribution of external potential only (-mgh). Show that the particle density follows the barometric height equation (1). 2.) For N particles in a open system at constant pressure p and temperature T, let there...
  22. G

    Statistical thermodynamics - mean energy of a nonlinear oscillator

    Homework Statement Consider a classical one-dimensional nonlinear oscillator whose energy is given by \epsilon=\frac{p^{2}}{2m}+ax^{4} where x,p, and m have their usual meanings; the paramater, a, is a constant a) If the oscillator is in equilibrium with a heat bath at temperature T...
  23. S

    Statistical thermodynamics- ideal gases mixture (Reif 3.6)

    A glass bulb contains air at room temperature and at a pressure of 1 atmosphere. It is placed in a chamber filled with helium gas at 1 atmosphere and at a room temperature. A few months later, the experimenter happens to read in a journal article that the particular glass of which the bulb is...
  24. R

    A challenging statistical thermodynamics problem.

    Homework Statement Consider the case of a gas in the atmosphere. Assume that the temperature is a constant. Based on the Maxwell Boltzmann distribution, at sea level the atmosphere contains 78.1% N2, 21% O2, 0.9% Argon, and 0.036 CO2. What are the ratios at the the top of Everest? (Molecular...
  25. H

    Statistical thermodynamics - system of oscillators

    Homework Statement A system of 10 oscillators, characterised by a \beta^ parameter of ln(3/2) is in equilibrium with a heat bath. Determine the probability that the system should possesses Q quanta.Homework Equations p(Q) proportional to e^(-Beta*Q)The Attempt at a Solution I have seen a...
  26. D

    Statistical thermodynamics: number of states of particle in central potential

    Homework Statement Give the number of states (energy of the state smaller than E<0) \Phi(E) of a spinless particle with mass m in the central potential V(\vec{r})=-\frac{a}{\left|\vec{r}\right|}. Homework Equations The Attempt at a Solution Hi, the hamiltonian of this problem...
  27. P

    Entropy of an ideal gas using statistical thermodynamics

    I cam across a question that asks for the entropy of an ideal gas of N molecules in which the energy of each molecule can assume 2 and only 2 distinct values 0 and E. It gives occupation numbers to represent the energy levels respectively with a fixed total energy. Now I know S=kln(omega) =...
  28. J

    Statistical thermodynamics question

    How do you use the C_ij matrix to find the harmonic frequency (or frequencies) of a diatomic molecule, the OH (hydroxyl) radical? (I have no idea how to set it up for this.) This is from Feynmann's book on Statistical Mechanics...
  29. A

    Statistical Thermodynamics - Help Wanted

    Statistical Thermodynamics - Help Wanted :( (My translation skills sucks, I hope it is understandable.) Three spins, placed at vertices of an equilateral triangle, are put in the external magnetic field with density B. Hamiltonian of the system: H = -J \sum_{<i,j>} s_{i} s_{j} - \gamma...
  30. D

    Thermodynamics and Statistical Thermodynamics References

    Hi, i'm an Applied Physics student (2nd year). Could you please tell me a book to study thermodynamics from? I've studied from Thermodynamics, Kinetic Theory, and Statistical Thermodynamics by Sears - Salinger and it is pretty good. Has anyone "tested" : 1) An Introduction to Thermal Physics by...
  31. J

    Seeking a Rigorous Treatment of Statistical Thermodynamics

    Hi all, I'm looking for a rigorous (in a mathematical sense) treatment of statistical thermodynamics. I'm at the tail end of a class on stat thermo that used the book by Bowley and Sanchez. This book is not what I'm looking for. Does anyone have any suggestions?
  32. C

    Phase Space Cells - Statistical Thermodynamics

    Homework Statement 1.) Explain why it is necessary to divide phase space into quantified cells of a finite size. 2.) Why is it necessary to know the size of these cells to over come the Gibb's paradox? Homework Equations The Attempt at a Solution 1.) I think it's something to...
  33. M

    Rigorous statistical thermodynamics?

    I took classical (engineering) thermodynamics a few years ago, and this semester I am taking a statistical thermodynamics class from the physics department. We are using the book "Fundamentals of statistical and thermal physics" by Reif, which seems to be a pretty good book. Unfortunately I...
  34. M

    Statistical Thermodynamics

    (1) Derive the expression for \muPG for a monatomic molecule (like argon) which has no internal structure and only has translational energy. where \muPG is the perect gas contribution to the chemical potential. (2)Treating the molecule N2 as a linear, rigid molecule is an excellent...
  35. A

    Question of Statistical Thermodynamics (Boltzmann Distribution)

    I apologize if this is the wrong thread but since this relates to thermo I figured this would be a good place to post this question. This is a problem that was assigned to us for physical chemistry but I can't find a good justification for one of the problems. 1. A system containing 38...
  36. T

    How Does Adding a Zero-Energy Particle Affect System Entropy?

    [SOLVED] Basic Statistical Thermodynamics Homework Statement Two distinguishable particles are to be distributed among nondegenerate energy levels 0,e,2e,3e... such that the total energy is U = 2e If a distinguishable particle with zero energy is added to the system show that the entropy...
  37. U

    Prof teaches Statistical thermodynamics in a Classical Thermodynamics class

    I'm just finishing up a "Classical Thermodynamics" class. Here is a list of topics we covered: Chapter 1: Ideal gas, equipartition of energy, heat and work, heat capacities, rates of processes Chapter 2: Multiplicity of an Einstein solid, of an ideal gas, of interacting systems Chapter 3...
  38. S

    Mathematica Mathematical formalism of classical and statistical thermodynamics

    Does anyone else have a lot of trouble comprehending the derivations in statistical mechanics? To me the mathematics feels somewhat archaic. Somehow it just seems as though it'd be neater if it was dealt with using matrix or operator methods. I always have trouble with the concept of entropy...
  39. M

    Statistical thermodynamics

    I think I get the concept of fermi-dirac and bose-einstein statistics can follow the derivation of their distribution functions as per Stastical physics by Guénault but I'm having severe trouble trying to apply them:redface:.For what I imagine is a simple question, two identical particles are...
  40. A

    Strange derivation: Statistical thermodynamics

    This isn't a homework question. I'm studying this from a book (Thermal Physics by Kittel & Kroemer) currently. Up til now I've had no problem following it. There's one derivation that's got me a little stumped however. I had thought my calculus was proficient enough, but I'm just not seeing...