Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Statistical mechanics and thermodynamics ↓


Advanced Thermodynamics


Introduction

Thermodynamics is the branch of physics that deals with heat, work, and temperature and their relation to energy and entropy. It forms the basis for understanding a variety of physical systems, from simple machines to complex chemical reactions. Advanced thermodynamics extends these principles by using concepts of statistical mechanics to analyze and predict the behavior of systems from a microscopic perspective.

Basic concepts

Systems and environments

In thermodynamics, we often talk about a "system" and its "surroundings." A system is any part of the universe we choose to focus on, and everything else is considered its surroundings. The boundary between the system and the surroundings can be physical or imaginary.

States and processes

The state of a thermodynamic system is defined by its macroscopic properties, such as temperature, pressure, and volume. A process is a change from one state to another, and it can be described quantitatively by changes in these properties.

Laws of thermodynamics

First law of thermodynamics

The first law of thermodynamics is the principle of energy conservation, which states that the total energy of an isolated system is constant. It can be expressed as:

ΔU = Q - W

where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the system.

Second law of thermodynamics

The second law of thermodynamics introduces the concept of entropy, which is a measure of disorder. It states that in any energy transfer or transformation, the total entropy of an isolated system can never decrease over time. This law implies that processes occur in a definite direction, not in the opposite direction.

Third law of thermodynamics

The third law of thermodynamics states that as the temperature of a system approaches absolute zero, the entropy of an ideal crystal approaches a constant minimum. This law helps determine the absolute entropy of substances.

Statistical mechanics and thermodynamics

Microstates and macrostates

Statistical mechanics provides a link between the microscopic properties of molecules and macroscopic thermodynamic quantities. A "microscopic state" is a specific configuration of the components of a system, while a "macroscopic state" is defined by macroscopic properties such as pressure and temperature. Several microscopic states can form a single macroscopic state.

Boltzmann's entropy formula

Ludwig Boltzmann introduced a statistical definition of entropy, relating the number of microstates, Ω , with the entropy, S , using the formula:

S = k_B * ln(Ω)

where k_B is the Boltzmann constant and ln is the natural logarithm.

Thermodynamic efficiency

Internal energy

Internal energy, denoted by 'U', is the total energy present in a system. It includes kinetic energy generated by the motion of particles and potential energy generated by interactions.

Enthalpy

Enthalpy, 'H', is a potential function useful in processes taking place at constant pressure. It is defined as:

H = U + PV

Where P is the pressure and V is the volume.

Helmholtz free energy

The Helmholtz free energy, 'A', measures the work that can be obtained from a closed system at constant volume and temperature. It is given as:

A = U - TS

where T is the temperature and S is the entropy.

Gibbs free energy

Gibbs free energy, 'G', is very useful in chemistry and biology for processes at constant pressure and temperature. It is calculated as follows:

G = H - TS

Visualization of thermodynamics concepts

PV diagram

The pressure-volume (PV) diagram is a commonly used graph in thermodynamics. The x-axis represents volume, and the y-axis represents pressure. The area under the curve in a PV diagram represents the work done by or on the system. Here is a simple example:


      
      
      P
      V
      Work

TS diagram

The temperature-entropy (TS) diagram is another useful diagram: the area under the curve represents the heat transfer during a process. Here is an abstract of the TS diagram:


      
      
      Tea
      S
      Heat

Applications in real life

Heat engine

Heat engines convert heat into work. The working material absorbs heat from the heat source, performs work as it moves in a cycle, and removes waste heat to a heat sink. A practical example of a heat engine is the internal combustion engine found in cars.

Refrigerator

Refrigerators use work input to transfer heat from a low-temperature region to a high-temperature region, effectively cooling the interior below ambient temperature. This process is essentially a reverse heat engine.

Phase transition

Understanding thermodynamics is important to explain phase transitions such as melting, freezing, and boiling. For example, at the boiling point, the liquid state changes to the gaseous state.

Summary

Advanced thermodynamics covers important concepts needed to understand energy transformations and physical properties in a variety of situations. Using the principles of both classical and statistical thermodynamics, you can more broadly predict system behaviors, which is important in a variety of scientific and engineering fields.


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Advanced Thermodynamics

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