What is Microstates: Definition and 51 Discussions

A microstate or ministate is a sovereign state having a very small population or very small land area, usually both. The meanings of "state" and "very small" are not well-defined in international law. Recent attempts, since 2010, to define microstates have focused on identifying political entities with unique qualitative features linked to their geographic or demographic limitations. According to a qualitative definition, microstates are "modern protected states, i.e. sovereign states that have been able to unilaterally depute certain attributes of sovereignty to larger powers in exchange for benign protection of their political and economic viability against their geographic or demographic constraints." In line with this and most other definitions, examples of microstates include Andorra, the Federated States of Micronesia, Liechtenstein, the Marshall Islands, Monaco, Palau, and San Marino. The smallest political entity recognized as a sovereign state is Vatican City, with around 1,000 citizens as of 2017 and an area of only 44 hectares (110 acres). Some microstates are city-states consisting of a single municipality.
Microstates are distinct from micronations, which are not recognized as sovereign states. Special territories without full sovereignty, such as the British Crown dependencies, the special administrative regions of China, and the overseas territories of various recognized states are also not usually considered microstates.

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  1. 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...
  2. 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...
  3. K

    A How Does Quantum Theory Provide a Measure for Microstates in Phase Space?

    *Pathria, Statistical mechanics*"The microstate of a given classical system, at any time, may be defined by specifying the instantaneous positions and momenta of all the particles constituting the system. Thus. If ##N## is the number of particles in the system, the definition of a microstate...
  4. K

    I A calculation of microstates

    *Pathria, Statistical mechanics pg 11,4ed* The text is long but it is straightforward. The question is at last about the equation given at end In order to find the number of microstates ##\Omega(N,V,E##) author writes " In other words, we have to determine the total number of (independent)...
  5. H

    Microstates for a gas of hard spheres in a box

    Hi, I have to found the number of microstates for a gas of N spheres of radius r and volume v in box taking into account the reduced volume after each sphere. V sphere << V box. I'm struggling to find the microstates in general. I don't see how to find the number of microstates without knowing...
  6. Kaguro

    A Meaning of density of microstates in phase space

    Hello all. I am studying stat mech from Pathria's book. It says a system is completely described by all positions and momenta of all the N particles. This maybe represented by a single point in 6N-D gamma space. So, each point is a (micro)state. Now if we restrict the system (N,V,E to E+ΔE)...
  7. L

    Determine how many microstates and macrostates (Thermodynamics)

    Since this is a two-state paramagnet where N = 40, therefore the microstate is ##40^2##? But I am not sure how to proceed to count the number of macrostates? Because from what I understand of what a macrostate is, shouldn't there a specific outcome to be stated in order to determine how many...
  8. E

    B What is the significance of Δx and Δp in the ideal gas entropy equation?

    Up to an undetermined constant ##a## the entropy of an ideal gas goes like$$S = k_B N\ln \left[ a^\frac{3}{2} \left(\frac{V}{N}\right) \left(\frac{E}{N}\right)^{\frac{3}{2}} \right]$$In some notes is written: And then they identify ##\Omega = \left(\frac{\Delta x \Delta p}{w}\right)^{3N}##...
  9. Clara Chung

    Question about microstates and electrons

    In 8a) I don't understand the question, does spin = 1/2 mean the magnitude of the spin or the z-component of the spin is 1/2? Can the electrons possesses be any of the spin=1/2 or -1/2? Are the electrons distinguishable?
  10. 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...
  11. L

    A Classical statistical physics -- Number of microstates

    Phase volume is it the same as the number of total microstates in some physical system? Phase volume= volume of phase space. Or there is some difference?
  12. C

    Microcanonical Ensemble

    Suppose that the number of microstates available to a system of N particles depends on N and the combined variable UV^2, where U is the internal energy and V is the volume of the system. The system initially has volume 2m^3 and energy 200 J. It undergoes an isentropic expansion to volume 4m^3...
  13. G

    How Do Microstates, kT, and Entropy Interact in Statistical Mechanics?

    Hi community, I'm trying to get my head round all these concepts. So entropy is given by S= k ln W. where W is the number of microstates of a system. Know let's imagine there is a box containing a number of gas atoms let's say the gas atoms have a current position and velocity and say you can...
  14. G

    Does Laplace's demon know about entropy?

    Hi. I read this thread with great interest and have similar question: In a deterministic universe, does entropy exist for Laplace's demon? Since he knows the universe to it's microstate, does the term "macrostate" even make sense to him? And say there is a "half-demon" that only knows the...
  15. Kara386

    Change in number of microstates

    Homework Statement An isolated macroscopic system at 300K absorbs a photon with ##\lambda = 550nm##. What is the relative increase ##\frac{\Delta \Omega}{\Omega}## in microstates. Homework EquationsThe Attempt at a Solution The energy of the photon is ##E = \frac{hc}{\lambda}## so that would...
  16. Kara386

    Number of microstates in multi-particle system

    Homework Statement Find the number of accessible microstates for a multi-particle system whose energy depends on temperature as ##U = aT^n## where a is a positive constant and ##n>1##. Use the fact that ##S = \int \frac{C_v}{T}dT## Homework EquationsThe Attempt at a Solution ##U = nC_vdT ##...
  17. U

    Calculating number of microstates to find entropy

    In the Boltzmann entropy formula , the number of microstates is calculated according to Maxwell-Boltzmann statistics , i.e. , W = n!/Πki! , Σki = n . Why cannot we use some other method , such as Bose-Einstein or Fermi-Dirac statistics ?
  18. 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...
  19. J

    Canonical Ensemble Microstates

    Hi, I've been looking at working with the canonical ensemble and getting the probabilities of a system being at a certain energy. For reference, I am following something of the form given under 'Canonical Ensemble' in this article...
  20. 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...
  21. P

    Macrostates and microstates

    Homework Statement A collection of three non-interacting particles shares 3 units of energy. Each particle is restricted to having an integral number of units of energy. a)How many macrostates are there? b)How many microstates are there in each of the macrostates? c)What is the probability of...
  22. Ascendant78

    Thermal Physics - energy, microstates, and probabilities

    Homework Statement Homework Equations The Attempt at a Solution The first part I'm not worried about, but the second part is worked out in the "relevant equations" section. Honestly, it looks like more magic than a Harry Potter movie going on there to me. I'm at a loss as to what...
  23. A

    Amount of microstates - phase space volume

    Homework Statement Dear all, I am desperately trying to solve the following exercise, but unfortunately can't find any resources how to properly calculate the phase space volume. Given is a system of ##N>>1## classical particles that are allowed to move in a cylinder with a Radius of ##R##...
  24. H

    Showing Rigid Rotor Microstates with Angular Momentum ≤ M

    Homework Statement By evaluating the "volume" of the relevant region of its phase space, show that the number of microstates available to a rigid rotor with angular momentum less or equal to M is (M/ħ)2. Homework Equations Consider the motion in the variables θ and φ, with...
  25. M

    Microstates of an Einstein solid

    My confusion isn't exactly with a homework problem, but more with an example that is key to understanding a homework problem. So I am posting here anyway. 1. Homework Statement The example is of an Einstein solid, with N=3 oscillators. The book lists the multiplicity of each macrostate, with...
  26. avito009

    Entropy as number of microstates

    As we know S=Q/T. And Entropy is defined as number of microstates of a system. So does that prove that, the lower the temperature the more the microstates available?
  27. T

    Exploring Microstates in an Isolated System of N Atoms with Three Energy States

    Homework Statement Consider an isolated (fixed total energy) system of N atoms each of which may exist in three states of energies: −e, 0, e. Let us specify the macrostates of the system by N, E (the total energy), and n (the number of atoms in the zero energy state). a) Identify explicitly...
  28. L

    Calculate number microstates? Why not include acceleration?

    I want to calculate the number of microstates in a system for, say, n indistinguishable particles in a 3D box. Some examples I see just represent one microstate as a list of positions. Other examples use a list of positions and a list of velocities (translational and rotational). And if...
  29. F

    [Statistical Physics] Microstates in a large system/Boltzmann entropy

    Homework Statement For a box containing 1m^{3} of nitrogen at S.T.P., estimate the number of microstates which make up the equilibrium macrostate. Homework Equations S = Nk_{b}(ln\frac{V}{N} + \frac{5}{2} + \frac{3}{2}ln\frac{2πmk_{b}T}{h^{2}}) where the entropy of a volume, V ...
  30. D

    Finding the number of microstates

    Hello, I don't understand how we can find the number of accessible microstate Homework Statement A system has 4 levels of energy, E1 E2 E3 & E4 with degeneracies of 1 , 3, 4 and 5. And we have E2=2E1, E3=3E1 .. We distribute 5 same particles on the different levels of energy...
  31. S

    Real microstates change within mixing two gases

    Hello, I'm looking for some changes of energy states of particles, as two gases are mixing together without total volume changing. That would explain increasing entropy better than their position combination. I know the particle energy states are getting more closely (the band gaps are reducing)...
  32. E

    Counting microstates in Boltzmanns principle

    Homework Statement Explain why the number of microstates W in Boltzmanns principle, is W = ƩNi! / ∏Ni! when i ideal gasses are mixed at constant volume and temperature. Ni is the number of particles of component i. Homework Equations S=klnW , where W is the number of microstates...
  33. T

    Partition function related to number of microstates

    Hi, I have a question about the partition function. It is defined as ## Z = \sum_{i} e^{-\beta \epsilon_{i}} ## where ##\epsilon_i## denotes the amount of energy transferred from the large system to the small system. By using the formula for the Shannon-entropy ##S = - k \sum_i P_i \log...
  34. O

    Calculate number of microstates of n harmonic oscillators

    Homework Statement Consider a system of N localized particles moving under the influence of a quantum, 1D, harmonic oscillator potential of frequency ω. The energy of the system is given by E=(1/2)N\hbarω + M\hbarω where M is the total number of quanta in the system. compute the total...
  35. P

    Are the number of microstates of a gas just equivalent to pressure

    I am quite confused about this area. First entropy does not contain any reference to volume. So if we can theoretically set the entropy of A and B gas samples as the same but in different volumes. If A is in a larger volume it would be able to exhibit a larger number of microstates? Yet the...
  36. A

    Microstates of two (similar looking) configuration

    Homework Statement Question: A box contains 1 mole of a gas. Consider two configuration : a) each half of the box contains half the molecules and b) each third of the box contains one-third of the molecules. Which configuration has more microstate?Homework Equations The only equations to be...
  37. E

    Firmi-Dirac Stats. - Calculate number of microstates.

    Homework Statement Determine the number of microstates where two particles have have energy 2eV, three have 3eV, three have energy 4eV, and two have 5eV. 10 indistinguishable Particles 1 particle allowed per state. ε1 = 1eV ε2 = 2eV ε3 = 3eV ε4 = 4eV ε5 = 5eV ε6 = 6eV g1...
  38. Y

    Relating temperature to energy in microstates of a system (Blundell)

    Hi there, I am working through the preliminary chapters of "Concepts in Thermal Physics" by Blundell. I have not really studied statistics before, so have a lot to get to grips with, but I thought I was making headway until this question cast questions over my understanding again! Homework...
  39. A

    Introductory Statistical Mechanics - counting number of microstates

    Homework Statement Consider a system composed of 2 harmonic oscillators with frequencies w and 2w respectively (w = omega). The total energy of the system is U=q * h_bar * w, where q is a positive negative integer, ie. q = {1, 3, 5, ...}. Write down the number of microstates of the system...
  40. B

    Statistical mechanics - microstates & entropy

    Homework Statement a) Derive an asymptotic expression for the number of ways in which a given energy E can be distributed among a set of N, one-dimensional harmonic oscillators, the energy eigenvalues of the oscillators being (N+\frac{1}{2})\hbar\omega, n=0, 1, 2, .... b)Find the corresponding...
  41. B

    The # of Microstates of a given Macrostate

    Homework Statement So this question has been bugging me because I can't begin to start it. The question is, prove that \Omega, the number of microstates of the combination of two physical states in thermal contact is a Gaussian of the energy of one of the states. \Omega is given here as...
  42. R

    Microstates & Entropy of two subsystems of a 4 particle system? (ensembles?)

    Homework Statement **************************************************************** A four-particle system is composed of 2 two–particle subsystems. Subsystem I has par- ticles A and B, which can have a maximum internal energy U(AB) = 4{E}. Subsystem II has particles C and D, in which the...
  43. S

    Definition of the number of microstates accessible to a system

    Hi guys, I'm going through Reif's book on statistical mechanics and have a question on the parameter omega, the number of microstates accessible to a system. Let's imagine a big box, and divide it into two equal halves. Now place a gas with a macroscopic amount of molecules in the left half, and...
  44. Y

    How Many Oscillators in a Carbon Nanoparticle with 5000 Atoms?

    Homework Statement A carbon nanoparticle (very small particle) contains 5000 carbon atoms. According to the Einstein model of a solid, how many oscillators are in this block? I didn't know how to start, can anyone give me a suggesstion??
  45. F

    Calculating Microstates and Oscillators in a Collection of Objects

    Microstates oscillators?? Homework Statement I have no idea where to begin on this problem. but here is what it asks Consider an object containing 6 one-dimensional oscillators (this object could represent a model of 2 atoms in an Einstein solid). There are 4 quanta of vibrational energy...
  46. Q

    Black hole microstates and string theory

    Can someone give me a Pop science explanation of what do strings and Branes have got to do with black hole entropy. The no of degrees of freedom or the no of microstates of the black hole corresponding to a given macrostate is given by exp(S),where S is entropy of black hole which is related to...
  47. J

    Statistical Mechanics (multiplicity/accesbile microstates question)

    I am trying to show that the change in number of accessible microstates, and therefore the change in the multiplicity function g of a simple system is g=\left( \frac{\tau_F^2}{\tau_1\tau_2}\right)^{\frac{mC_V}{k_B}} where the system is two identical copper blocks at fundamental...
  48. K

    Thermal equilibrium of A and B, microstates and quanta

    Homework Statement System A consisting of 10^23 oscillators for which hw=10^-20 J, is in thermal contact with system B, consisting of 2x10^23 similar oscillators. The joint system has internal energy 3x10^3 J. Calculate the number of quanta in A and B, and the temperature, when A and B are...
  49. K

    Microstates and oscillators help

    Homework Statement The number of microstates of a system of N oscillators containing Q quanta of energy homework is given by W(N,Q) = (N+Q-1)!/[(N-1)!Q!] Show that when one further quantum is added to the system the number of microstates increases by a factor of approximately (1+N/Q)...
  50. Q

    Microstates and multi-dimensional oscillators

    Consider an object containing 9 one-dimensional oscillators (this object could represent a model of 3 atoms in an Einstein solid). There are 5 quanta of vibrational energy in the object. (a) How many microstates are there, all with the same energy? 1287 microstates (b) If you examined a...
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