PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Statistical mechanics and thermodynamics → Statistical mechanics ↓


Microstates and Macrostates


Statistical mechanics and thermodynamics are complex topics in physics that help us understand and predict how systems of particles behave. One of the most important concepts in this field is the idea of microstates and macrostates. These concepts form the basis of much of the work we do in statistical mechanics.

Understanding microstates

Let's start by discussing microstates. In statistical mechanics, a microstate represents a specific detailed configuration of a system. It includes the exact position and momentum of each particle in the system.

To help understand this, imagine a simple system consisting of two dice. Each dice can show a value from 1 to 6. In this case, a microstate would be any unique combination of the numbers shown on the two dice. For example, a microstate could be (3, 5) where the first dice shows 3 and the second dice shows 5.

Visual example of microstates for two dice

1,1 | 1,2 | 1,3 | 1,4 | 1,5 | 1,6 2,1 | 2,2 | 2,3 | 2,4 | 2,5 | 2,6 3,1 | 3,2 | 3,3 | 3,4 | 3,5 | 3,6 4,1 | 4,2 | 4,3 | 4,4 | 4,5 | 4,6 5,1 | 5,2 | 5,3 | 5,4 | 5,5 | 5,6 6,1 | 6,2 | 6,3 | 6,4 | 6,5 | 6,6
1,1 | 1,2 | 1,3 | 1,4 | 1,5 | 1,6 2,1 | 2,2 | 2,3 | 2,4 | 2,5 | 2,6 3,1 | 3,2 | 3,3 | 3,4 | 3,5 | 3,6 4,1 | 4,2 | 4,3 | 4,4 | 4,5 | 4,6 5,1 | 5,2 | 5,3 | 5,4 | 5,5 | 5,6 6,1 | 6,2 | 6,3 | 6,4 | 6,5 | 6,6

Each row and column in the above table represents a unique microscopic state of this dice system, resulting in a total of 36 possible microscopic states for two 6-sided dice.

Understanding macrostates

Microstates give us detailed information, while macrostates help us understand the system in a more generalized form. Macrostates are defined by the macroscopic properties of the system, such as temperature, pressure, volume, or total energy.

Continuing the example of dice, let's define a macrostate as the sum of the values on two dice. If we are only interested in the total sum, many different microstates can give the same macrostate. For example, if the macrostate has a sum equal to 7, it can be achieved by many microstates such as (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).

Visual example of macrostates for the sum of two dice

Sum = 2: (1,1) Sum = 3: (1,2), (2,1) Sum = 4: (1,3), (2,2), (3,1) Sum = 5: (1,4), (2,3), (3,2), (4,1) Sum = 6: (1,5), (2,4), (3,3), (4,2), (5,1) Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) Sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2) Sum = 9: (3,6), (4,5), (5,4), (6,3) Sum = 10: (4,6), (5,5), (6,4) Sum = 11: (5,6), (6,5) Sum = 12: (6,6)
Sum = 2: (1,1) Sum = 3: (1,2), (2,1) Sum = 4: (1,3), (2,2), (3,1) Sum = 5: (1,4), (2,3), (3,2), (4,1) Sum = 6: (1,5), (2,4), (3,3), (4,2), (5,1) Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) Sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2) Sum = 9: (3,6), (4,5), (5,4), (6,3) Sum = 10: (4,6), (5,5), (6,4) Sum = 11: (5,6), (6,5) Sum = 12: (6,6)

Each sum represents a different macrostate, and as shown, each macrostate can contain multiple microstates. The macrostate with a sum of 7 is particularly interesting because it has the most microstates (6 microstates). This makes it statistically the most likely macrostate if you roll two dice randomly.

Importance of microstates and macrostates

In statistical mechanics, the main relationship between microstates and macrostates is fundamental. A macrostate with more associated microstates is more likely to occur because there are more ways to obtain it. This concept is important when discussing entropy, which is a measure of the number of microstates associated with a given macrostate.

Entropy can be understood with this formula:

S = k_B * ln(Ω)
S = k_B * ln(Ω)

where S is the entropy, k_B is the Boltzmann constant, and Ω is the number of microstates corresponding to a macrostate. Entropy is often associated with disorder, but in statistical mechanics, it is seen as a measure of how many ways a system can be arranged at the microscopic level while still looking the same at the macroscopic level.

Example of calculating entropy

Imagine a very simple physical system that has a total of 4 possible microstates: (A), (B), (C), and (D), and two macrostates defined as M1 = {(A), (B)} and M2 = {(C), (D)}. If each microstate is equally probable, the entropy of these macrostates can be calculated as follows:

For macrostate M1:

Ω1 = 2 S1 = k_B * ln(2)
Ω1 = 2 S1 = k_B * ln(2)

For macrostate M2:

Ω2 = 2 S2 = k_B * ln(2)
Ω2 = 2 S2 = k_B * ln(2)

The entropy of both macroscopic states is the same because they have the same number of microscopic states.

Real world application of microstates and macrostates

In real systems, especially systems made up of a large number of particles such as a gas in a container, the number of microstates is astronomically large, making direct calculation impossible. Instead, statistical mechanics allows us to predict the macroscopic properties of these systems by focusing on the macrostates.

For example, consider a gas enclosed in a container. Instead of tracking the position and speed of every molecule (which is a nearly impossible task), we consider the macrostates of the gas defined by its temperature, pressure, and volume. We estimate the most probable behavior of the gas by considering the most probable macrostate.

PV = nRT
PV = nRT

This is the formula for the ideal gas law, which relates the macroscopic quantities pressure P, volume V, amount of substance n, and temperature T for a gas, and explains it in terms of the microscopic representation.

Microscopic states and macroscopic states in thermodynamics

In thermodynamics, the concepts of microscopic states and macroscopic states help us understand heat and work, important concepts that describe how energy is transferred within a system. The second law of thermodynamics, which states that the entropy of an isolated system never decreases, can be interpreted in terms of microscopic states and macroscopic states because systems naturally tend toward macroscopic states with the most entropy.

This explains why, for example, heat flows from hotter to colder objects. The system evolves into a macrostate where the energy is spread more evenly among the available microstates, effectively increasing the total entropy of the system.

Conclusion

The interaction between microscopic states and macroscopic states is at the heart of statistical mechanics and thermodynamics. It transforms our understanding from a deterministic Newtonian framework to a probabilistic framework that accommodates the complexity of real-world systems. By recognizing that a macroscopic state with more probable microscopic states is statistically more probable, we gain profound insights into the natural trends observed in thermodynamic systems, such as the increase of entropy or the approach to equilibrium.

Whether you're considering a simple pair of dice or the vastness of a room filled with gas, the concepts of microstates and macrostates are powerful tools for understanding the behavior of systems made up of many interacting components. By focusing on these macroscopic details, we can make accurate predictions even without knowing the details of every particle involved.


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