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 ↓


Phase transition


In the fields of statistical mechanics and thermodynamics, phase transitions play an important role in understanding the behavior of matter. These transitions describe changes between different states of matter, such as solid, liquid, and gas, and also include more exotic states. Phase transitions are characterized by abrupt changes in physical properties at certain critical points. Understanding these transitions involves combining thermodynamic principles and statistical mechanics, which provides insight into the microscopic interactions that give rise to macroscopic phenomena.

What is the phase?

Phase refers to a region of space in which all of the physical properties of a substance are essentially the same. Common examples of phases include solids, liquids, and gases. In solids, atoms or molecules are arranged in a structured, rigid lattice. Liquids have a definite volume but take the shape of their container, and gases fill the entire volume available to them.

Solid: Particles are closely packed in a fixed arrangement. Liquid: Particles are close together but can move past each other. Gas: Particles are far apart and move freely.

Types of phase transitions

Phase transitions can generally be classified into different types based on the change in physical properties:

  • First-order phase transition: It involves latent heat (energy change) and discontinuous change in properties. Melting, boiling and condensation are examples.
  • Second-order phase transitions: These do not involve latent heat but are continuous derivatives of these, such as the transition in magnetic materials known as the Curie point.

First-order phase transition

First-order phase transitions are characterized by a discontinuous change in the first derivative of the free energy with respect to some thermodynamic variable. Examples include:

Example 1: Melting and Freezing

When ice melts, it absorbs heat, causing a change in enthalpy without a change in temperature—an example of a first-order transition.

H2O (solid, ice) → H2O (liquid, water)

Example 2: Boiling

The boiling of water is another first-order transition, where the temperature remains constant as the liquid changes into vapor, absorbing heat.

H2O (liquid) → H2O (gas, steam)

Second-order phase transitions

In a second-order or continuous phase transition there is no latent heat and no discontinuity in the first derivative. Instead, higher derivatives of the free energy show discontinuities.

Example: Ferromagnetism and Curie Point

In ferromagnetic materials, at a certain temperature known as the Curie point, the material loses its permanent magnetism. This transition occurs smoothly without the involvement of latent heat.

T < TC : Material is ferromagnetic. T > TC : Material becomes paramagnetic.

Critical points and universality

At the critical point, the properties of the system change dramatically. Critical points indicate conditions of temperature, pressure, or composition at which phase boundaries disappear. Near the critical point, systems often exhibit universality—the critical exponents and scaling functions are similar in different substances.

Van der Waals equation and critical point

The van der Waals equation for real gases provides a classic example of critical behavior:

(P + a(n/V)2)(V/n - b) = RT

At the critical point, the derivative of pressure with respect to volume is zero:

(dP/dV)T = (d2P/dV2)T = 0

These conditions reveal important constants such as critical temperature, pressure, and volume, which are unique to each substance.

Order parameters

The order parameter is a measure that describes the degree of order in a phase transition. In ferromagnetic materials, the magnetization serves as the order parameter:

M (magnetization) = 0 above TC M ≠ 0 below TC

During the transition from liquid to gas, the density difference between the phases can serve as an order parameter.

Mean field theory

Mean field theory provides an approximate way to study phase transitions by treating the effect of all other molecules on any one molecule as a uniform field. In the context of ferromagnetism, mean field theory predicts the behavior of a magnetic material near its Curie point.

H = Hext + αM

where H is the magnetic field seen by a spin and α is a measure of the average field contribution.

Ising model

The Ising model is a mathematical model used in statistical mechanics in physics to understand ferromagnetism. It consists of discrete variables called spins that can be in one of two states (+1 or -1). The model studies the interactions between spins on a lattice.

One-dimensional Ising model

The simplest form involves spins arranged in a line, with interactions between nearest neighbors:

H = -J ∑<i,j> Si Sj - h ∑ Si

Graphical representation

Understanding phase transitions often benefits from a visual representation. Below is a simplified illustration of a common phase diagram, showing the different phases of a substance in pressure-temperature space.

Pressure temperature Solid liquid Gas

Applications and implications

Phase transitions have important applications and implications in various fields of science and technology. In materials science, understanding phase transitions aids in the development of new materials and alloys. In cosmology, phase transitions have been hypothesized to play a role in the evolution of the early universe, such as during times such as the inflationary epoch.

By understanding phase transitions, scientists and engineers can create and manipulate materials with specific properties, contributing to advances in technology and industry.

Concluding remarks

The study of phase transitions is fundamental to the field of physics, which connects the macroscopic and microscopic worlds. It provides a window into the complex interactions that determine the states of matter and enables advances in many scientific fields. With ongoing research, the exploration of phase transitions continues to unravel the mysteries of the universe and physical properties.


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Phase transition

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