8.2 Non-equilibrium Thermodynamics and Onsager Reciprocal Relations

Non-equilibrium thermodynamics explores systems with net flows of matter or energy. It's all about understanding how things change when they're not in balance, using concepts like entropy production rate and phenomenological equations.

The Onsager reciprocal relations are a big deal here. They show how different processes in a system are connected, like how heat and electricity can influence each other. This helps us make sense of complex systems and design better technologies.

Non-equilibrium Thermodynamics

Fundamental Concepts

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• Non-equilibrium thermodynamics studies systems that are not in thermodynamic equilibrium, meaning there are net flows of matter or energy
• The entropy production rate represents the rate at which entropy is generated within the system due to irreversible processes and is a key quantity in non-equilibrium thermodynamics
• Linear non-equilibrium thermodynamics assumes that the system is close to equilibrium, and the flows are linearly related to the thermodynamic forces, which are gradients of intensive variables (temperature, concentration, etc.)
• The phenomenological equations in linear non-equilibrium thermodynamics relate the flows to the forces through phenomenological coefficients, which are material properties

Onsager Reciprocal Relations

• The Onsager reciprocal relations state that the matrix of phenomenological coefficients is symmetric, reducing the number of independent coefficients
• These relations are derived from the principle of microscopic reversibility, which states that the probability of a microscopic process and its time-reversed process are equal at equilibrium
• The derivation involves expressing the entropy production rate in terms of the flows and forces and using the principle of microscopic reversibility to show that the cross-coefficients are equal

Onsager Reciprocal Relations

Interpretation and Implications

• The Onsager relations imply that a gradient in one thermodynamic variable can cause a flow in another variable, known as cross-effects or coupled processes
• Examples of coupled processes include:
  • Thermoelectric effect: coupling between heat and electric current
  • Diffusion-viscous flow coupling: coupling between concentration gradient and viscous flow
• The Onsager relations lead to a deeper understanding of the symmetry and interconnectedness of transport processes in non-equilibrium systems

Derivation

• The derivation of the Onsager reciprocal relations starts with expressing the entropy production rate in terms of the flows and forces
• The principle of microscopic reversibility is then applied to show that the cross-coefficients in the phenomenological equations are equal
• This derivation establishes a fundamental symmetry in the linear response of non-equilibrium systems to thermodynamic forces

Onsager Coefficients and Transport

Relation to Transport Properties

• The diagonal Onsager coefficients are related to the transport properties of the system, such as electrical conductivity, thermal conductivity, and diffusion coefficients
• The off-diagonal Onsager coefficients represent the coupling between different transport processes and are related to cross-effects like the Seebeck coefficient in thermoelectrics
• These relations allow for the experimental determination of Onsager coefficients through measurements of transport properties

Fluctuation-Dissipation Theorem

• The Onsager coefficients can also be related to the fluctuations of the corresponding extensive variables through the fluctuation-dissipation theorem
• The fluctuation-dissipation theorem states that the response of a system to a small perturbation is related to the fluctuations of the system at equilibrium
• The Onsager coefficients can be expressed in terms of the time correlations of the fluctuations of the corresponding extensive variables (energy, particle number, etc.)
• This theorem provides a microscopic basis for the Onsager relations and highlights the connection between fluctuations and dissipation in non-equilibrium systems

Coupled Transport Processes

Problem-Solving Approach

• The phenomenological equations with the Onsager relations can be used to set up and solve problems involving coupled transport processes
• The problem-solving steps include:
  1. Identify the relevant thermodynamic forces and flows in the system
  2. Express the entropy production rate in terms of these variables
  3. Write the phenomenological equations using the Onsager coefficients, relating the flows to the forces
  4. Apply the Onsager reciprocal relations to simplify the equations by equating the cross-coefficients
  5. Use boundary conditions and conservation laws to solve for the unknown variables (temperature profile, electric current density, etc.)

Applications

• The results obtained from solving coupled transport problems can be used to:
  • Calculate the efficiency of energy conversion devices like thermoelectric generators
  • Analyze the transport properties of materials (semiconductors, nanomaterials, etc.)
• Examples of applications include:
  • Optimizing thermoelectric materials for waste heat recovery
  • Designing solid-state cooling devices based on the Peltier effect
  • Studying the transport of ions and molecules in biological membranes
• Understanding and harnessing coupled transport processes is crucial for developing efficient and sustainable energy technologies