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 ↓


Legendre transforms and thermodynamic potentials


Introduction

Legendre transforms and thermodynamic potentials are important concepts in the study of advanced thermodynamics, particularly within statistical mechanics. These mathematical tools allow transformations between different sets of variables, providing a deeper understanding of various thermodynamic processes. Thermodynamic potentials provide insight into the behavior of systems, especially when transformations occur at constant volume, temperature, pressure, or chemical potential.

Understanding Legendre transforms

Legendre transforms are mathematical techniques used to switch from one set of variables to another in optimization problems. They are named after the mathematician Adrien-Marie Legendre. In the context of thermodynamics, these transformations enable conversion between different thermodynamic potentials.

Consider a function f(x). The Legendre transform of this function with respect to x is defined as:

    l(y) = x*y - f(x)

Here, y is the derivative of f relative to x; y = df(x)/dx. The new function L(y) now uses y as the independent variable instead of x.

Visual example of Legendre transforms

Suppose we have a curve that represents the function f(x) To visualize the Legendre transform, imagine drawing tangents at different points along this curve. Each tangent can be represented by its slope and y-intercept. The collection of these tangent points can then be used to create the Legendre transform.

Examples in physics

Consider the internal energy U(S, V), where S is the entropy and V is the volume. The differential form is given as:

    dU = TDS - PDV

Here, T is the temperature and P is the pressure. Using the Legendre transform, we can define new potentials, such as the Helmholtz free energy F and the Gibbs free energy G

Thermodynamic efficiency

Thermodynamic potentials are properties or state functions that are used to measure the "free" energy within a system. They help describe the energy available to do work in various situations. The main thermodynamic potentials are:

  • Internal energy U
  • Helmholtz free energy F = U - TS
  • Gibbs free energy G = U - TS + PV
  • Enthalpy H = U + PV

Internal energy

Internal energy U is the total energy present in a system. It takes into account both potential and kinetic energy at the microscopic level. The change in internal energy is represented by the equation:

    dU = TDS - PDV

This differential form shows how the internal energy changes with respect to entropy and volume.

Helmholtz free energy

The Helmholtz free energy F is particularly useful when dealing with systems at constant temperature and volume. It is defined as:

    F = U – TS

The Helmholtz free energy indicates the maximum work that can be obtained from a closed system at constant volume.

Gibbs free energy

The Gibbs free energy G is often used in chemical reactions and processes that occur at constant pressure and temperature. It is given as:

    G = U – TS + PV

Gibbs free energy measures the maximum useful work that can be obtained from a closed system.

Enthalpy

Enthalpy H is a measure of the total heat content in a system. It is essential for understanding heat transfer processes and is defined as:

    H = U + PV

Enthalpy is particularly useful for systems undergoing constant pressure processes.

Applications in thermodynamics

Thermodynamic potentials are important in predicting and understanding the equilibrium states of various thermodynamic systems.

Example 1: Gibbs free energy in chemical reactions

Gibbs free energy helps determine whether a chemical reaction will proceed spontaneously or not. A negative change in Gibbs free energy (ΔG < 0) indicates a spontaneous reaction, while a positive change (ΔG > 0) indicates nonspontaneousness.

Example 2: Enthalpy in a phase change

Enthalpy plays an important role in understanding phase changes such as melting and boiling. During these processes, the pressure remains constant, making enthalpy an essential quantity for evaluating heat transfer.

Visualization of thermodynamic potential

Internal Energy (U) Helmholtz free energy (F) Gibbs free energy (G) Enthalpy (H)

Conclusion

Legendre transforms and thermodynamic potentials are powerful tools for exploring and understanding various thermodynamic processes and systems. They offer the flexibility to switch between different thermodynamic variables, providing insightful insights into the underlying behavior of materials and reactions under different conditions.


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Legendre transforms and thermodynamic potentials

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