Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Thermodynamics ↓


Statistical mechanics


Statistical mechanics is a branch of physics that attempts to explain and predict the properties of macroscopic systems based on the known behavior of their microscopic components. The field bridges the gap between quantum mechanics and thermodynamics. It uses statistical methods to relate the microscopic properties of individual atoms and molecules to the macroscopic, observable properties of substances.

Basic concepts

To understand statistical mechanics, we first need to clarify several fundamental concepts, including states, groups, and probabilities.

States

In statistical mechanics the "state" of a system defines its particular microscopic state, including all the microscopic details such as the position and momentum of each particle. However, in practice it is often impossible to know all these details precisely.

For example, consider a box containing many gas particles. Each particle may be in many different microscopic states due to variations in position and velocity. It may be impossible to track them all.

Each circle in the SVG above represents a different particle in a gas. The particles are constantly moving and colliding, causing their microscopic states to change regularly.

Ensembles

An ensemble is a collection of virtual copies of a system, each of which represents a possible state the real system could be in. Several types of ensembles are used in statistical mechanics, such as microcanonical ensembles, canonical ensembles, and grand canonical ensembles. They are used to model systems with different constraints (e.g., energy, particles).

  • Microcanonical ensemble: An isolated system with fixed energy, volume, and number of particles.
  • Canonical ensemble: a closed system in thermal equilibrium with a heat bath at a fixed temperature.
  • Grand canonical ensemble: an open system that can exchange both energy and particles with its environment.

Possibilities

In statistical mechanics, systems are described in terms of probability. Each possible state (microstate) of a system has a probability. The probability distribution over these microstates allows the calculation of average quantities, such as average energy or pressure.

From micro to macro

Statistical mechanics explains how thermodynamic properties (such as temperature, pressure, and entropy) arise from the behavior of microscopic components (atoms and molecules). Using physical laws and statistical methods, it predicts the collective behavior of large numbers of particles.

Boltzmann distribution

One of the key results in statistical mechanics is the Boltzmann distribution. It provides the probability P(E) that a system in thermal equilibrium will be in a state with energy E :

P(E) = (1/Z) * exp(-E/kT)

where Z is the partition function, k is the Boltzmann constant, and T is the temperature. The Boltzmann factor exp(-E/kT) indicates that higher-energy states are less probable than lower-energy states.

Partition function

The partition function Z is an important concept. It is the sum of all possible situations and it normalizes the probabilities to ensure that they sum to one:

Z = Σ exp(-E_i/kT)

The partition function is central because it connects the microscopic properties of the system to its macroscopic properties. From Z, quantities such as internal energy U, free energy F, entropy S, pressure P, and more can be derived.

Entropy

In statistical mechanics, entropy is a measure of the number of distinct ways to arrange a system, usually interpreted as a measure of disorder. Mathematically, the entropy S can be expressed using the probabilities of the microstates p_i :

S = -k Σ p_i log(p_i)

Example: Ideal gas

Consider an ideal gas, a simple model where the gas particles do not interact in any way other than elastic collisions. Its behavior can be described using statistical mechanics.

For an ideal gas in a canonical group, distributions and properties such as the internal energy U and entropy S can be calculated. For example, the average energy per particle in a gas is directly proportional to the temperature:

U = (3/2) * N * k * T

where N is the number of particles. This result matches what has been found using classical thermodynamics.

The role of statistical mechanics in thermodynamics

Statistical mechanics provides a microscopic basis for the macroscopic laws of thermodynamics. It explains how thermodynamic processes behave by modeling the aggregate behavior of large numbers of particles.

Emergence of thermodynamic laws

Key thermodynamic principles such as the laws of thermodynamics emerge naturally from statistical descriptions. For example, the second law of thermodynamics, which states that entropy tends to increase, can be viewed as a consequence of the probabilistic behavior of particles tending toward high-entropy configurations over time.

First law of thermodynamics

The first law, which deals with the principle of energy conservation, can be viewed from a microscopic perspective in terms of the momentum and energy of individual particles in the system. Energy changes in the macroscopic system can be attributed to changes in particle behavior and interactions.

Applications of statistical mechanics

Statistical mechanics plays an important role in a variety of scientific fields beyond pure physics. Here are some examples:

Condensed matter physics

Statistical mechanics is the basis for the study of condensed matter, such as solids and liquids. It provides information about phase transitions (e.g., from liquid to solid), superconductivity, and other phenomena.

Molecular and chemical physics

It helps to understand molecular dynamics and reaction kinetics. For example, reaction rates and equilibria can be predicted by considering the statistical behavior of molecules.

Biophysics and biomolecules

Statistical mechanics is used to model complex biological systems, such as protein folding. The way proteins find their stable configuration among many possibilities can be understood through statistical approaches.

Conclusion

Statistical mechanics is a powerful framework that explains how the collective behavior of microscopic components in a system gives rise to macroscopic phenomena. Using statistical tools and principles, it provides a deeper understanding of heat, work, and related concepts, linking the microscopic and macroscopic worlds within physics.


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Statistical mechanics

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