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 ↓


Onsager interpersonal relations


The Onsager reciprocal relation is an essential concept in thermodynamics and statistical mechanics, especially when considering non-equilibrium systems. Proposed by Lars Onsager, who won the Nobel Prize in Chemistry in 1968, this principle provides insight into the linear response of thermodynamic systems near equilibrium. Let us discuss this topic more as we understand its significance in a deeper way.

Introduction to disequilibrium thermodynamics

In equilibrium thermodynamics, systems are typically in a state where macroscopic properties do not change over time. However, many real-world processes are far from equilibrium, where systems experience gradients such as temperature, pressure, or chemical potential. Non-equilibrium thermodynamics aims to describe these processes.

Consider an isolated system where there are certain flows (such as heat or matter) and forces (such as gradients). These forces drive the system away from equilibrium, producing measurable flows.

Basic principles

Onsager's approach was to look at the forces and flows (fluxes) in a system. He recognized that the flux in a system can be represented as a linear combination of all the forces applied to the system:

    j i = Σ L ij x j

Here, J i denotes fluxes (e.g., thermal fluxes, particle fluxes), X j denotes thermodynamic forces (e.g., gradients in temperature or chemical potential), and L ij are phenomenological coefficients that relate them.

Onsager's theory of interpersonal relations

Onsager's reciprocal relations state that the phenomenological coefficients are symmetric in a system near equilibrium:

    L i j = L g

This symmetry implies that the effect of the force X j on the flux J i is equal to the effect of the force X i on the flux J j.

Relation with microscopic reversibility

The reciprocal relationships emerge from the principle of microscopic reversibility, which states that the microscopic laws of physics are reversible in time. This reversibility is a key concept in statistical mechanics.

Mathematical illustration

Consider a system with two types of fluxes, such as heat and particle fluxes. Suppose we have:

    J 1 = L 11 x 1 + L 12 x 2
    
    J 2 = L 21 x 1 + L 22 x 2

Here, J 1 can be the heat flux, and J 2 can be the particle flux. According to Onsager's relations, L 12 = L 21.

Example: thermoelectric effect

In a thermoelectric device, a temperature gradient can induce an electric current (Seebeck effect), and vice versa, an electric current can cause a temperature gradient (Peltier effect). These effects can be described as follows:

        J q = L qq ΔT + L qi ΔV
        
        J E = L EQ ΔT + L EE ΔV

Here, J q is the heat flux, J e is the electric current density, ΔT is the temperature difference, and ΔV is the voltage difference. Onsager's reciprocity states that L qe = L eq, which reflects the symmetric effect of electric fields and temperature gradients.

Visualizing the Onsager relations

Consider a system where two processes (A and B) influence each other. Let us represent the processes as follows:

    A ↔ B

Process A Process B L 12 L 21

This simple representation shows how processes can influence each other, and emphasizes the symmetric relationship defined by Onsager.

Applications and significance

Onsager's reciprocal relations have significant implications in various fields of science, including physics, chemistry, and even biology, because of their universality in describing non-equilibrium systems.

Chemical reactions

Consider a chemical reaction in a closed system where two or more reactants affect each other's rates. Onsager's relation can predict how a change in the concentration of one reactant might affect the flow of another.

Biological systems

In biological cells, transport processes such as ion exchange and nutrient absorption can be described using Onsager's principles. The flow of ions and molecules through membranes can affect other transport processes, which are governed by the same reciprocal relationships.

Conclusion

Onsager reciprocal relations provide a fundamental understanding of the behavior of non-equilibrium systems. By revealing the underlying symmetry in transport processes, these relations enable scientists and engineers to efficiently predict system behavior in a variety of scenarios.


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Onsager interpersonal relations

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