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 ↓


Ising model


The Ising model is an important concept in statistical mechanics, particularly in the study of phase transitions and critical phenomena. Originally proposed as a simplified model of ferromagnetism, the Ising model has proven to be a powerful abstraction for explaining complex behavior in many physical systems. This detailed explanation provides information about the Ising model, its mathematical formulation, applications, and relation to real-world phenomena.

Introduction to the Ising model

In physics, understanding changes in the state of matter is a central question. Whether it is ice melting into water or a magnet losing its magnetism above a certain temperature, these changes are known as phase transitions. These phenomena can be described using the Ising model, which captures the essence of these changes at a fundamental level. The Ising model is named after Ernst Ising, who studied it as part of his doctoral research in 1925.

The essence of the Ising model lies in its simplicity. It considers a regular lattice, which can be thought of as a grid of points, where each point represents an atomic "spin". These spins can take one of two values, usually +1 or -1. The interaction between neighbouring spins defines the energy of the system, which in turn determines its behaviour at different temperatures.

Mathematical formulation of the Ising model

The Ising model is mathematically formulated as follows:

H = -J ∑ SᵢSⱼ - h ∑ Sᵢ

Here, H denotes the Hamiltonian or the total energy of the system. The sum of SᵢSⱼ represents the nearest-neighbor interactions, where J is the interaction strength. h is an external magnetic field, and Sᵢ term can take values of either +1 or -1.

Explanation of terms

  • Spin (Sᵢ): Individual units that can be +1 or -1, representing the magnetic moment.
  • Interaction strength (J): a parameter that measures the interaction between neighboring spins.
  • External magnetic field (h): An applied field that can affect the overall alignment of the spins.

Now, let's take a deeper look at how the Ising model is used to study phase transitions. Phase transitions occur at a critical temperature, where the material changes from one phase to another. For ferromagnetic systems, this is the temperature at which the system transitions from overall magnetization to pure magnetization.

Phase transitions and critical phenomena

The study of phase transitions through the Ising model focuses on how the spins interact and align with each other. These interactions depend heavily on temperature:

  • Low Temperatures: At low temperatures, the spins align, maximizing the number of parallel neighbors, leading to a magnetic state. This is due to the tendency of the system to minimize its energy.
  • High temperatures: As temperatures rise, thermal fluctuations disrupt the alignment, leading to a disordered, non-magnetic state.

The critical temperature, known as the Curie temperature in ferromagnetic systems, is the location where the phase transition occurs, and the Ising model helps to understand this criticality.

Visual representation

To understand the Ising model, consider a simple case of a one-dimensional lattice:

+1 -1 +1

Here, each circle represents a spin on a lattice. Red circles represent spins with a +1 value, while blue circles represent spins with a -1 value. The line connecting them represents nearest-neighbor interactions.

Two-dimensional lattice

A two-dimensional lattice can be visualized as follows. Consider a 3x3 grid where each point represents a spin:

Each circle represents a spin that can be in one of two states: +1 (red) or -1 (blue). Nearest-neighbor interactions are vertical and horizontal between adjacent spins.

Real-world applications

Although the Ising model was initially developed to study ferromagnetism, its applications extend to many other areas. Here are some notable examples:

  • Phase transitions: Beyond magnetism, the Ising model helps understand vapor-liquid transitions, alloy formation, and more.
  • Neuroscience: The Ising model is used to model neural networks, where brain neurons are compared to spins with a binary state.
  • Network theory: used to model social networks, where individual preferences (analogous to spins) depend on interactions with neighbors.
  • Biology: Understanding DNA transcription and other biological processes through spin-like models.

Conclusion

The Ising model remains a cornerstone of theoretical physics, demonstrating the power of simple models in explaining complex phenomena. It captures the essential features of phase transitions, paving the way for research in a variety of fields. Its appealing simplicity, yet profound implications, make it the subject of widespread study and application.

Studying the Ising model provides a framework not only for understanding physical systems but also for understanding a wide variety of phenomena that share the feature of cooperative behavior. It beautifully demonstrates how macroscopic phenomena arise from simple, local interactions, a core principle in the study of complex systems.


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Ising model

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