"In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities."
The study of the behavior of large systems of particles, and the relationships between their macroscopic properties and the properties of their individual particles.
Probability and statistics: Basic tools for understanding the behavior of large systems characterized by an enormous number of variables.
Thermodynamics: Laws of thermodynamics that describe the behavior of macroscopic systems in terms of thermodynamic variables like temperature, pressure, volume, and energy.
Kinetic theory of gases: Describes the behavior of a gas in terms of the motion of the particles that compose it.
Boltzmann distribution: The probability distribution that describes the distribution of energies of a large number of particles in thermal equilibrium at a certain temperature.
Partition function: The partition function is a mathematical function used in statistical mechanics. The partition function is a sum over all possible quantum states of the system in question, and is used to calculate thermodynamic properties such as free energy, entropy, and specific heat.
Ensembles: A collection of micro-states that describe a macro-state of a system. The three most important ensembles are the canonical ensemble, the grand canonical ensemble, and the microcanonical ensemble.
Ideal gases: Used as a simple model for real gases, the ideal gas assumes that molecules are point-like, have no volume, and do not interact with each other.
Statistical thermodynamics: An approach to thermodynamics that is based on the laws of probability and statistics, which provides a microscopic understanding of macroscopic thermodynamic properties.
Phase transitions: The transition from one phase of matter to another. Examples include the transition from a solid to a liquid or from a liquid to a gas.
Quantum mechanics: The laws of physics that govern the behavior of particles at the atomic and subatomic levels.
Entropy: A measure of the disorder or randomness of a system. In statistical mechanics, entropy is defined in terms of the probability theory.
Chemical equilibrium: The state of a chemical reaction when the concentrations of products and reactants no longer change with time.
Molecular dynamics: A simulation technique used to understand the behavior of large molecular systems by solving the equations of motion for each molecule.
Thermodynamic fluctuations: Thermal fluctuations arise from the random thermal motion of molecules, and are responsible for fluctuations in thermodynamic quantities like pressure, volume, or energy.
Monte Carlo simulations: A computer-based simulation technique that involves generating random numbers that are used to simulate molecular systems.
Brownian motion: The random, erratic movement of microscopic particles due to the random motion of molecules in a fluid.
Lattice models: Simplified models of complex systems in which the molecules are arranged in a regular lattice.
Non-equilibrium statistical mechanics: The study of systems that are far from equilibrium and do not obey the laws of thermodynamics.
Transport phenomena: The study of flow, diffusion, and heat transfer in fluids or solids.
Fluctuation-dissipation theorem: A fundamental result that relates the response of a system to an external perturbation to the fluctuations in the equilibrium state of the system.
Classical Statistical Mechanics: Classical statistical mechanics deals with the properties of macroscopic systems composed of many particles that follow classical mechanics, such as atoms and molecules. It utilizes the concepts of thermodynamics, probability theory, and statistical physics to study the thermodynamic properties of these systems.
Quantum Statistical Mechanics: Quantum statistical mechanics is a branch of statistical mechanics that deals with the properties of quantum systems. It uses the principles of quantum mechanics, such as wave-particle duality and uncertainty principles, to predict the thermodynamic properties of quantum systems.
Non-equilibrium Statistical Mechanics: Non-equilibrium statistical mechanics is a branch of statistical mechanics that deals with the study of systems that are not in thermodynamic equilibrium. It is concerned with the dynamics of systems that are driven away from equilibrium, and the application of statistical methods to the analysis of their behavior.
Finite Size Scaling: Finite size scaling is a technique used to study the thermodynamic properties of systems of finite size. It is based on the idea that the behavior of a finite size system can be related to the behavior of an infinite size system through scaling relations.
Stochastic Processes: Stochastic processes are mathematical models for systems that have a random component. They are used to describe the behavior of a wide range of systems, from molecular systems to financial markets.
Monte Carlo Methods: Monte Carlo methods are a class of computational algorithms that use random sampling to solve complex problems. They are used extensively in statistical mechanics to simulate the behavior of complex systems.
Density Functional Theory: Density functional theory is a computational method used in statistical mechanics to calculate the energy of a system of interacting particles. It is based on the idea that the energy of a system can be determined by the density of the particles in the system.
Renormalization Group Theory: The renormalization group theory is a set of mathematical methods used in statistical mechanics to study the behavior of systems near critical points. It is used to study the phase transitions in condensed matter systems.
"It explains the macroscopic behavior of nature from the behavior of such ensembles."
"Its applications include many problems in the fields of physics, biology, chemistry, and neuroscience."
"Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion."
"Statistical mechanics arose out of the development of classical thermodynamics."
"Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates."
"James Clerk Maxwell, who developed models of probability distribution of such states."
"Josiah Willard Gibbs, who coined the name of the field in 1884."
"Non-equilibrium statistical mechanics focuses on the issues of microscopically modeling the speed of irreversible processes that are driven by imbalances."
"Examples of such processes include chemical reactions and flows of particles and heat."
"The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles."
"It applies statistical methods and probability theory."
"It does not assume or postulate any natural laws."
"It explains the macroscopic behavior of nature from the behavior of such ensembles."
"Classical thermodynamics is primarily concerned with thermodynamic equilibrium."
"Microscopic parameters fluctuate about average values and are characterized by probability distributions."
"It clarifies the properties of matter in aggregate, in terms of physical laws governing atomic motion."
"Physics, biology, chemistry, and neuroscience."
"Microscopically modeling the speed of irreversible processes that are driven by imbalances."
"Ludwig Boltzmann, James Clerk Maxwell, and Josiah Willard Gibbs."