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 ↓


Renormalization group theory


Renormalization group (RG) theory is a fundamental framework in statistical mechanics and thermodynamics that helps us understand phase transitions. It began as a way to handle problems in quantum field theory, but its applications extend to a wide variety of physical systems, especially in understanding how small-scale interactions can affect large-scale phenomena.

Understanding phase transitions

To understand the concept of renormalization and its importance, we must first discuss phase transitions. Phase transitions occur when a substance changes its phase, such as ice melting and turning into water or water evaporating into steam. Generally, phase transitions involve changes in temperature and pressure. Common examples include:

  • Solid to Liquid: At 0°C, ice melts into water.
  • Liquid to Gas: At 100°C water boils and turns into steam.

These changes are not just limited to matter. These changes also occur in magnetism when a material changes from magnetically ordered (ferromagnetic) to disordered (paramagnetic), which happens at its critical temperature called the Curie point.

The need for renormalization

Renormalization occurs because physical systems can exhibit behavior at different scales that cannot be easily understood by looking only at the microscopic or macroscopic scales. Let us understand this concept with a simple example:

Imagine you are tasked with describing a forest. - At a microscopic level, you focus on leaves, insects, and bark textures. - At a macroscopic level, you look at the forest's shape, the number of trees, and the ecosystem as a whole. How do you connect these two levels?

Systems undergoing phase transitions, such as magnets at their Curie temperature, behave similarly. Near critical points, small fluctuations at the microscopic level can lead to very large changes macroscopically. This is because the system is 'scale-invariant'; the way it behaves does not depend on the level of detail we choose to describe it.

The basic idea of reparameterization group theory

The main idea behind renormalization is to understand how physical systems behave when observed at different scales. Here is a visualization of this concept:

Microscopic mapsIntermediate mapMacroscopic maps

The process of renormalization involves "coarse-graining," which means systematically removing microscopic details to study the properties that remain relevant at larger scales. These remaining properties are known as renormalized parameters.

Fixed points

In the context of renormalization, a fixed point is a set of parameters that remain unchanged when the renormalization procedure is applied. These points often correspond to phase transition points or critical points. Understanding the behavior of the system when approaching these fixed points reveals much about the nature of the phase transition.

Example: Ising model

One of the most famous models in statistical mechanics, the Ising model, provides a clear context for applying RG theory. The Ising model describes a ferromagnet as a lattice of spins that can point up or down. Interactions occur only between neighbouring spins.

Consider a simple 2D Ising model on a square lattice: - Each spin S i,j can take values of +1 or -1. - The Hamiltonian of the system is given by: H = -J ∑ S i,j S i+1,j - J ∑ S i,j S i,j+1 where J is the coupling constant between neighboring spins.

To apply RG theory to this problem, one will consider increasing the length scale by grouping blocks of spins together into single spins and determining the effective interactions between these larger blocks.

The goal is to find fixed points of such transformations. At a fixed point, the properties of the system itself are identical, which shows the critical nature of the phase transition.

Critical exponents

In the study of phase transitions, critical exponents describe how physical quantities diverge as they approach the critical point. Renormalization group theory provides a systematic way to calculate these exponents. For example, consider the magnetization M:

Near the critical temperature T c, the spontaneous magnetization behaves as: M ∝ (T - T c) β Here, β is a critical exponent that characterizes this behavior. RG techniques allow us to connect β to the fixed point properties.

Contributions of renormalization group theory

RG theory revolutionized the way physicists understand phase transitions and critical phenomena. Through its ability to handle scale invariance and universality beautifully, RG theory provides the following insights:

  • Universality: Subtly different systems can exhibit the same important behaviour, if they have the same symmetries and dimensions.
  • Scaling rules: Critical exponents follow specific scaling relationships that simplify their calculation and understanding.

Applications beyond physics

Renormalization techniques extend beyond traditional physics applications to the following areas:

  • Biology: Understanding the robustness of biological systems to external fluctuations.
  • Finance: Analyzing market behavior and sudden crashes that resemble phase transitions.

Conclusion

Renormalization group theory is an essential framework that has had a profound impact on physics and beyond. By understanding how systems behave under changes in scale, physicists can reveal the universal principles that guide the complex behavior of matter under phase transitions.

The study of reparameterization groups not only provides an intellectual understanding of important phenomena but also provides practical methods for calculating important properties such as critical exponents and scaling laws, shaping the modern theoretical landscape of phase transition studies.


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