Computational Physics
Introduction to Statistical Mechanics
Statistical Mechanics is the branch of physics that uses probability theory to study the behavior of systems of a large number of particles. It connects macroscopic thermodynamic properties with microscopic particle interactions.
Classical Statistical Mechanics
Classical Statistical Mechanics deals with systems of classical particles and describes their behavior using the principles of classical mechanics and statistics.
Quantum Statistical Mechanics
Quantum Statistical Mechanics extends classical principles to systems where quantum effects are significant. It describes systems of particles following Fermi-Dirac or Bose-Einstein statistics.
The Canonical Ensemble
The Canonical Ensemble represents a system in thermal equilibrium with a heat reservoir at a fixed temperature. It is characterized by the partition function, which encodes the statistical properties of the system.
The Grand Canonical Ensemble
The Grand Canonical Ensemble is used to describe systems in contact with a heat and particle reservoir. It allows the exchange of energy and particles and is characterized by temperature, volume, and chemical potential.
Microcanonical Ensemble
The Microcanonical Ensemble describes an isolated system with fixed energy, volume, and particle number. It is useful for studying systems with fixed energy and exploring entropy and thermodynamic quantities.
Phase Transitions and Critical Phenomena
Phase Transitions and Critical Phenomena study changes between different states of matter, such as solid, liquid, and gas. Critical phenomena occur near phase transition points where physical properties show power-law behavior.
The Ising Model 1D
The Ising Model 1D is a mathematical model of ferromagnetism in statistical mechanics. It consists of discrete variables representing magnetic dipole moments of atomic spins, arranged in a linear chain.
The Ising Model 2D
The Ising Model 2D extends the 1D Ising model to two dimensions. It is used to study phase transitions, particularly in the context of magnetism, and exhibits a phase transition at finite temperature.
The Ising Model 3D
The Ising Model 3D further extends the Ising model to three dimensions. It is more complex and is used to study critical phenomena and phase transitions in three-dimensional systems.
The XY Model
The XY Model is a statistical model of spins on a lattice where each spin can rotate freely in a plane. It is used to study phase transitions, particularly the Berezinskii-Kosterlitz-Thouless transition.
The Heisenberg Model
The Heisenberg Model describes interactions between neighboring spins in a crystal lattice. It is used to study magnetic properties and phase transitions in materials.
The Potts Model
The Potts Model generalizes the Ising model to more than two possible spin states. It is used to study phase transitions and critical phenomena in systems with multiple states.
The q-state Potts Model
The q-state Potts Model extends the Potts model to q possible states for each spin. It is used to study a variety of phenomena in statistical mechanics, including magnetism and percolation.
The Blume-Capel Model
The Blume-Capel Model is a spin-1 Ising model that includes a single-ion anisotropy term. It is used to study tricritical points and phase transitions in magnetic systems.
The Ashkin-Teller Model
The Ashkin-Teller Model is a generalization of the Ising model that includes interactions between four spins. It is used to study complex phase transitions and critical behavior.
The Clock Model
The Clock Model is a variation of the Potts model where spins can take on discrete angles. It is used to study phase transitions and critical phenomena in systems with angular variables.
The Percolation Model
The Percolation Model studies the movement and connectivity of particles on a lattice. It is used to study phenomena like fluid flow in porous media and network robustness.
Spin Glasses
Spin Glasses are disordered magnetic systems with competing interactions. They are used to study complex energy landscapes and slow dynamics in disordered systems.
Lattice Gas Models
Lattice Gas Models describe particles distributed on a lattice with nearest-neighbor interactions. They are used to study fluid dynamics and phase transitions in fluids.
Cellular Automata Models
Cellular Automata Models consist of a grid of cells that evolve according to simple rules. They are used to study complex systems and emergent behavior from simple interactions.
Hard Sphere Models
Hard Sphere Models describe particles as hard spheres that cannot overlap. They are used to study phase transitions and thermodynamic properties of dense fluids and solids.
Soft Sphere Models
Soft Sphere Models describe particles that interact with soft, repulsive potentials. They are used to study the behavior of colloids, polymers, and other soft matter systems.
Lennard-Jones Potential Models
Lennard-Jones Potential Models describe particles interacting through the Lennard-Jones potential. They are used to study molecular dynamics, phase transitions, and properties of liquids and gases.
The Ideal Gas Model
The Ideal Gas Model describes a gas of non-interacting particles. It is used to study thermodynamic properties and provides a simple approximation for real gases.
The Van der Waals Gas Model
The Van der Waals Gas Model improves upon the ideal gas model by including interactions between particles. It is used to study real gases and phase transitions.
Real Gas Models
Real Gas Models describe gases with complex interactions between particles. They are used to study deviations from ideal behavior and properties of real gases.
The Bose-Einstein Condensate
The Bose-Einstein Condensate is a state of matter formed by bosons cooled to near absolute zero. It is used to study quantum phenomena on a macroscopic scale.
The Fermion System
The Fermion System describes particles that obey Fermi-Dirac statistics. It is used to study properties of systems like electrons in metals and nucleons in atomic nuclei.
The Boson System
The Boson System describes particles that obey Bose-Einstein statistics. It is used to study properties of systems like photons, phonons, and superfluid helium.
The Many-body System
The Many-body System studies interactions between a large number of particles. It is used to understand collective behavior and emergent properties in complex systems.
Quantum Monte Carlo Methods
Quantum Monte Carlo Methods are computational techniques used to study quantum systems. They are used to calculate properties of quantum systems with high accuracy.
Classical Monte Carlo Methods
Classical Monte Carlo Methods are computational techniques used to study classical systems. They are used to simulate thermodynamic properties and phase transitions.
Path Integral Monte Carlo
Path Integral Monte Carlo is a quantum Monte Carlo method that uses Feynman's path integral formulation. It is used to study quantum systems at finite temperatures.
Replica Monte Carlo
Replica Monte Carlo is a method used to study disordered systems like spin glasses. It uses multiple replicas of the system to sample the energy landscape.
Cluster Algorithms
Cluster Algorithms are Monte Carlo methods that group spins into clusters to improve efficiency. They are used to study phase transitions and critical phenomena.
Metropolis-Hastings Algorithm
The Metropolis-Hastings Algorithm is a Markov Chain Monte Carlo method used to sample from probability distributions. It is widely used in statistical physics and Bayesian statistics.
Heat Bath Algorithm
The Heat Bath Algorithm is a Monte Carlo method that simulates thermal baths. It is used to sample spin configurations in models like the Ising and Potts models.
Wang-Landau Algorithm
The Wang-Landau Algorithm is a Monte Carlo method that calculates the density of states. It is used to study systems with complex energy landscapes and phase transitions.
Multicanonical Ensemble
The Multicanonical Ensemble is a Monte Carlo method that samples from a flat energy distribution. It is used to study phase transitions and systems with rare events.
Quantum Spin Models
Quantum Spin Models describe interactions between quantum spins in a lattice. They are used to study magnetic properties and quantum phase transitions.
Hubbard Model
The Hubbard Model describes electrons in a lattice with on-site interactions. It is used to study strongly correlated electron systems and high-temperature superconductivity.
t-J Model
The t-J Model is a variation of the Hubbard model that includes exchange interactions. It is used to study high-temperature superconductivity and strongly correlated systems.
The Anderson Model
The Anderson Model describes localized electron states in a disordered lattice. It is used to study localization and transport properties in disordered systems.
Density Functional Theory
Density Functional Theory is a computational method used to study the electronic structure of matter. It is widely used in condensed matter physics, chemistry, and materials science.
Quantum Lattice Models
Quantum Lattice Models describe particles on a lattice with quantum mechanical interactions. They are used to study properties of materials and phase transitions.
Lattice Gauge Theory
Lattice Gauge Theory is a computational method used to study gauge theories on a lattice. It is used to study quantum chromodynamics and other field theories.
Molecular Dynamics Simulations
Molecular Dynamics Simulations are computational methods used to study the motion of particles. They are used to study properties of liquids, solids, and biological systems.
Langevin Dynamics
Langevin Dynamics is a method used to simulate the motion of particles with friction and random forces. It is used to study systems with thermal noise and dissipative dynamics.
Brownian Motion
Brownian Motion describes the random motion of particles in a fluid. It is used to study diffusion, transport properties, and thermal fluctuations in condensed matter systems.
Non-equilibrium Statistical Mechanics
Non-equilibrium Statistical Mechanics studies systems that are not in thermal equilibrium. It is used to study transport properties, reaction kinetics, and systems with external driving forces.
Renormalization Group Theory
Renormalization Group Theory is a mathematical framework used to study changes in a physical system as it is viewed at different length scales. It is used to study critical phenomena and phase transitions.
Mean Field Theory
Mean Field Theory is an approximation method used to study phase transitions and critical phenomena. It assumes that each particle in a system feels an average field due to all other particles.
Finite Size Scaling
Finite Size Scaling is a method used to study the behavior of systems near critical points by analyzing the effects of finite system size. It is used to extract critical exponents and phase transition properties.
Quantum Phase Transitions
Quantum Phase Transitions are transitions between different quantum states of matter at zero temperature. They are driven by quantum fluctuations and are studied using models like the quantum Ising model.
Topological Phases of Matter
Topological Phases of Matter are states of matter characterized by topological properties rather than local order parameters. They are studied using models like the Kitaev model and are important in understanding quantum computing.
Frustrated Systems
Frustrated Systems are systems in which competing interactions prevent the system from reaching a simple ground state. They are studied using models like the spin ice and kagome lattice.
Quantum Magnetism
Quantum Magnetism studies the magnetic properties of quantum systems. It is used to understand phenomena like magnetic ordering, spin waves, and quantum critical points.
High-temperature Superconductors
High-temperature Superconductors are materials that exhibit superconductivity at relatively high temperatures. They are studied using models like the Hubbard and t-J models to understand their mechanisms.
Quantum Hall Effect
The Quantum Hall Effect is a quantum phenomenon observed in 2D electron systems under strong magnetic fields. It is characterized by quantized Hall conductance and is studied using models like the Landau levels.
Anderson Localization
Anderson Localization is the absence of diffusion of waves in a disordered medium. It is studied using models like the Anderson model and is important in understanding electronic properties of disordered systems.
Quantum Monte Carlo in Condensed Matter
Quantum Monte Carlo in Condensed Matter uses Monte Carlo techniques to study quantum systems in condensed matter physics. It is used to calculate properties of systems like the Hubbard model and spin systems.
Quantum Dots and Wires
Quantum Dots and Wires are low-dimensional semiconductor structures that exhibit quantum confinement effects. They are studied using models like the effective mass approximation and are important in nanotechnology.
Spintronics
Spintronics studies the spin of electrons in solid-state systems and its applications in electronic devices. It is used to develop technologies like spin transistors and magnetic memory.
Nonequilibrium Green’s Functions
Nonequilibrium Green’s Functions are used to study transport properties and nonequilibrium phenomena in quantum systems. They are used to analyze systems driven out of equilibrium by external fields.
Monte Carlo Methods for Lattice Field Theory
Monte Carlo Methods for Lattice Field Theory are computational techniques used to study quantum field theories on a lattice. They are used to calculate properties of systems like quantum chromodynamics.
Quantum Monte Carlo for Bosonic Systems
Quantum Monte Carlo for Bosonic Systems uses Monte Carlo techniques to study systems of bosons. It is used to calculate properties of systems like superfluid helium and Bose-Einstein condensates.
Quantum Monte Carlo for Fermionic Systems
Quantum Monte Carlo for Fermionic Systems uses Monte Carlo techniques to study systems of fermions. It is used to calculate properties of systems like electrons in metals and nucleons in atomic nuclei.