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 ↓


Partition function


The concept of partition function is central to the field of statistical mechanics, which in turn is an essential part of thermodynamics in physics. This concept helps us connect the microscopic world of atoms and molecules to the macroscopic world of physics phenomena such as pressure, temperature and volume. By understanding the partition function, we can understand various properties of systems made up of a large number of particles such as gases, solids and liquids.

What is partition function?

In statistical mechanics, the partition function is a way of relating all possible states of a system in a way that incorporates their energies and the temperature of the system. It is usually denoted by the symbol Z and is defined for a system in thermal equilibrium at temperature T The partition function provides an essential connection between the microscopic states of a system and its macroscopic thermodynamic properties.

Mathematical definition

For a system with discrete energy levels, the canonical partition function is defined as:

Z = Σ e -E i /kT

Here:

  • E i i the energy of the ith state.
  • k is the Boltzmann constant.
  • T is the absolute temperature.
  • This total is over all states i

For systems with a constant energy level, the partition function is written as an integral:

Z = ∫ e -E/kT g(E) dE

Here g(E) is the density of states, which tells us how many states have a particular energy.

Why is the partition function important?

The partition function is a powerful tool because once we know it, we can calculate many macroscopic properties of the system. These include internal energy, free energy, entropy, and other thermodynamic quantities.

Relation to thermodynamic properties

Let us see how the partition function helps us obtain several thermodynamic quantities:

  • Internal Energy (U): The average energy of the system can be found as follows:
    U = -∂(ln(Z))/∂β
    where β = 1/kT.
  • Free energy (F): The Helmholtz free energy is:
    F = -kT ln(Z)
  • Entropy (S): Entropy can be obtained from:
    S = k (ln(Z) + βU)
  • Pressure (P): Pressure is obtained as the derivative of free energy with respect to volume:
    P = -∂F/∂V

Visual example: a two-tier system

To understand the partition function, let's consider a simple example: a system with only two energy levels. Let the energies of the two levels be E 0 = 0 and E 1 = ε.

The partition function Z for this system is:

Z = e -0/kT + e -ε/kT = 1 + e -ε/kT

The SVG representation of this system would look like this:

e 0 = 0 e 1 = ε

This example shows the sum of the exponential terms corresponding to each energy level, weighted by the Boltzmann factor, which determines the probability of the system being in a particular state.

Lesson example: an ideal gas

Consider an ideal gas, which is a group of non-interacting particles enclosed in a container. For an ideal gas, each particle can be in different states with different energy levels.

The partition function for a single ideal gas particle in three dimensions is given by:

Z = V/h 3 ∫∫∫ e -(p 2 /2m)/kT d 3 p

Where V is the volume of the container, h is Planck's constant, p is the momentum of the particle, and m is the mass of the particle.

Solving this integral, we get:

Z = (VkT/2πħ)

This expression highlights how the partition function increases with the volume of the container and the temperature.

Combination of multiple particles

For a collection of N differentiated particles, the total partition function is simply a product of the single-particle partition functions:

Z total = Z N

If the particles are indistinguishable, we must call it N!

Z total = Z N /N!

This distinction is important for accurately describing real world systems, especially high density gases.

Conclusion

The partition function is a central concept in statistical mechanics that provides a powerful link between the microscopic states of a thermodynamic system and its macroscopic properties. By understanding the partition function, physicists can derive important properties such as energy, entropy, and pressure, which provide insight into the behavior of various physical systems.

With examples ranging from simple systems with two energy levels to more complex systems such as ideal gases, the partition function provides a comprehensive framework that is essential to the study of statistical mechanics and thermodynamics.


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Partition function

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