MATE 664 Lecture 02

Introduction to Irreversible Thermodynaics

Author

Dr. Tian Tian

Published

January 7, 2026

Note

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Recap of Lecture 01 (Big Picture)

Key ideas from last lecture:

Thermodynamics tells us whether a process can occur
Kinetics tells us how fast it occurs
Equilibrium depends on time scales, not perfection
Free energy change $\Delta G^{0}$ determines spontaneity
Activation barrier $\Delta G^*$ controls rates

Recap: equilibrium is a time-scale concept

Equilibrium when

\[\tau_{\text{observation}} \gg \tau_{\text{relaxation}}\]

Many materials are metastable, not unstable
Diamond vs graphite is a kinetic story, not thermodynamic failure

Recap: entropy revisited

Entropy is not disorder
Entropy measures number of accessible microstates
Statistical view:

\[S = k_B \ln\!\Omega\]

Learning outcomes

After this lecture, you will be able to:

Describe why classical thermodynamics is insufficient for real kinetic processes
Define entropy production and irreversible processes
Relate macroscopic fluxes to thermodynamic driving forces
Apply Onsager’s linear response framework
Interpret irreversible thermodynamics from a microscopic viewpoint

Crash course on statistical mechanics (1)

Part of the introductions are adapted from MAT E 374 by Dr. Hao Zhang

Dynamics from Newton & Laplace point of view:

Transition from state A to B is deterministic
We just need to know $\{\mathbf{x}_i, \mathbf{v}_i, F_{ij}\}$

Crash course on statistical mechanics (2)

Why can’t we determine the system state using Newtonian Dynamics (yet)?

Astronomical numbers of microstates
Limitation of computational power
Waste of computational power

What is the alternative?

Deterministic -> probabilistic
How much computation is enough?

Crash course on statistical mechanics (3)

Let’s reintroduce the microscopic picture by Boltzmann

Crash course on statistical mechanics (4)

Microstate: a state in the phase space $\{\mathbf{x}_i, \mathbf{p}_i\}$
Macrostate: a specification of macroscopic constraints (e.g. $<E, V, N>$) that corresponds to many microstates
Ensemble: a set of constraints like $<E, V, N>$ describing the system

Central principle linking macrostate and microstates:

The observed macroscopic thermodynamic property $A_{obs}$ is an ensemble average of all possible microstates $i$

\[ A_{obs} = \langle A \rangle = \sum_i A_i \, p_i \]

Crash course on statistical mechanics (5)

Consider a canonical ensemble (constant $N, V, U$) ¹

Inner system (I): $U_I$, $\Omega(U_I)$
Reservoir system (R): $U_R$, $\Omega(U_R)$
Total system (universe, T): $U_T = U_I + U_R$, $\Omega(U_T) = \Omega(U_I) \times \Omega(U_R)$
$U_R \gg U_I$

I and R can only exchange internal energy, not particles or volume.

Q: The total internal energy of the universe $U_T$ is fixed. How do the inner and reservoir systems balance their energies?

Crash course on statistical mechanics (6)

Let’s first see the equilibrium state.

Assume at e.q., we have $U_I = U_I^{0}$
Probability of inner energy follows how many states in the reservoir they can exchange with:

\[P(U_I) \propto \Omega(U_T - U_I^0)\]

Since $U_T \approx U_R \gg U_I^0$, we can use Taylor series for $\Omega(U_T - U_I^0)$:

\[ \ln\!\Omega(U_T - U_I^0) = \ln\!\Omega(U_T) - \frac{\partial \ln\!\Omega}{\partial U} \bigg\vert_{U=U_T} \cdot U_I^0 + \dots \]

Crash course on statistical mechanics (7)

We know the Boltzmann equation $S = k_B \ln\!\Omega$

\[ \ln P(U_I^0) \propto \ln\!\Omega(U_T) - \frac{\partial S_T}{k_B \partial U} \bigg\vert_{U=U_T} \cdot U_I^0 \]

Recall the classical thermodynamic relation:

\[ dU = TdS - pdV \]

We have $\dfrac{\partial S}{\partial U}\bigg\vert_{V} = \dfrac{1}{T}$, therefore

\[ P(U_I^0) \propto \Omega(U_T) \exp(-\dfrac{U_I^0}{k_B T}) \]

This leads the famous Boltzmann distribution

Crash course on statistical mechanics (8)

What if we’re not at equilibrium?

$T_I$ and $T_R$ may be two distinct values.

We will apply local equilibrium assumption here
One postulate for stat. mech. is that the entropy of the universe always increases.
We go further by say the equilibrium maximizes the universe entropy

\[ \left( \dfrac{ \partial \ln \Omega(U_T) }{\partial U_I} \right)_{N, V, U} = 0 \]

Can we make sure it is the maximum?
What is the sign of $\partial^2 S/\partial U^2$?

Crash course on statistical mechanics (9)

\[ \left( \dfrac{ \partial \ln \Omega(U_I) }{\partial U_I} \right)_{N, V, U} + \left( \dfrac{ \partial \ln \Omega(U_R) }{\partial U_I} \right)_{N, V, U} = 0 \]

Use the relation $\partial U_I = -\partial U_R$

\[ \left( \dfrac{ \partial \ln \Omega(U_I) }{\partial U_I} \right)_{N, V, U} = \left( \dfrac{ \partial \ln \Omega(U_R) }{\partial U_R} \right)_{N, V, U} \]

Apply Boltzmann equation

\[ T_I^{-1} = T_R^{-1} \]

Yay! This is something we know, temperature at equilibrium.

What have we observed so far?

Probability distribution at equilibrium follows Boltzmann distribution (for canonical ensemble)
Equilibrium is a consequence of maximizing entropy of the universe
We can locally apply the equilibrium thermodynamics to describe kinetic process (though we don’t know how fast yet)

Formal framework

In Onsager’s Irreversible Thermodynamics, for any thermodynamic parameter $X$, we define

Affinity $F$: the influence of $X_i$ to the universe entropy \[ F_i = \frac{\partial S}{\partial X_i} \]
Flux $J$: the change of parameter $X_i$ (or density of it) per unit time \[ J_i = \frac{d X_i}{d t} \]
Kinetic Coefficient $L_{ij}$: influence of flux type $i$ by affinity type $j$ (linear term). \[ L_{ij} = \dfrac{\partial J_i}{\partial F_j} \]

Postulate 1: local equilibrium

It is generally possible to use principles of the continuum limit to define meaningful, useful, local values of various thermodynamics quantities, e.g., chemical potential $\mu_i$.
Useful kinetic theories can be developed by assuming a functional relationship between the rate of a process and the local departure from equilibrium (“driving force”).

Postulate 2: entropy production

Irreversible Process is An Increase of Entropy in the Universe

\[ \frac{dS}{dt} = \sum_i \frac{\partial S}{\partial X_i} \frac{\partial X_i}{\partial t} = \sum_i F_i J_i \]

Entropy can be created in the universe
Entropy can flow between micro-systems
Rate of entropy production (entropy generation + flow) must be positive

Postulate 3: linearity

The relation between $J$ and $X$ can be linearly described (resistive system).
Flux is only dependent on the affinity at the same time

\[\begin{align} J_{1} &= L_{11} F_{1} + L_{12}F{2} + \cdots + L_{1\Sigma}F_{\Sigma}\\ \vdots &\\ J_{\Sigma} &= L_{\Sigma1} F_{1} + L_{\Sigma2}F{2} + \cdots + L_{\Sigma\Sigma}F_{\Sigma}\\ \end{align}\]

Or in vector term \[ \mathbf{J} = \mathbf{L}\cdot\mathbf{F} \]

Coefficient $L_{ii}$ is called direct kinetic coefficient
- Affinity $F_{i}$ is called conjugate driving force w.r.t. $J_i$
Coefficient $L_{ij}(i\neq j)$ is called coupling coefficient

Postulate 4: reciprocity

Onsager proposed that

\[ L_{ij} = L_{ji} \]

This is a key component in material kinetics!

Summary

Irreversible processes are driven by thermodynamic forces
Each force generates a corresponding flux
Near equilibrium, fluxes depend linearly on forces
Empirical transport laws (Fourier, Ohm, Fick) are special cases of: \[ J_i = \sum_j L_{ij} F_j \]
Onsager reciprocity enforces symmetry: \[ L_{ij} = L_{ji} \]

Footnotes

Adapted from Green, Kinetics Transport, and Structure in Hard and Soft Materials, Ch 1↩︎

--- title: "MATE 664 Lecture 02" subtitle: "Introduction to Irreversible Thermodynaics" author: "Dr. Tian Tian" date: "2026-01-07" format: html: {} revealjs: output-file: slides.html pdf: output-file: L02.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L02.pdf) - Handwritten notes 👉 [Open in pdf](./public/L02_annotated.pdf) ::: ::: ## Recap of Lecture 01 (Big Picture) {.center} Key ideas from last lecture: - Thermodynamics tells us **whether** a process can occur - Kinetics tells us how **fast** it occurs - Equilibrium depends on time scales, not perfection - Free energy change $\Delta G^{0}$ determines spontaneity - Activation barrier $\Delta G^*$ controls rates ## Recap: equilibrium is a time-scale concept {.center} - Equilibrium when $$\tau_{\text{observation}} \gg \tau_{\text{relaxation}}$$ - Many materials are metastable, not unstable - Diamond vs graphite is a kinetic story, not thermodynamic failure ## Recap: entropy revisited {.center} - Entropy is **not disorder** - Entropy measures number of accessible **microstates** - Statistical view: $$S = k_B \ln\!\Omega$$ ## Learning outcomes {.center} After this lecture, you will be able to: - **Describe** why classical thermodynamics is insufficient for real kinetic processes - **Define** entropy production and irreversible processes - **Relate** macroscopic fluxes to thermodynamic driving forces - **Apply** Onsager’s linear response framework - **Interpret** irreversible thermodynamics from a microscopic viewpoint ## Crash course on statistical mechanics (1) *Part of the introductions are adapted from MAT E 374 by Dr. Hao Zhang* Dynamics from Newton & Laplace point of view: - Transition from state A to B is **deterministic** - We just need to know $\{\mathbf{x}_i, \mathbf{v}_i, F_{ij}\}$ ![Newtonian dynamics in a nutshell](./public/newtonian.png){width=600} ## Crash course on statistical mechanics (2) Why can't we determine the system state using Newtonian Dynamics (yet)? - Astronomical numbers of microstates - Limitation of computational power - Waste of computational power What is the alternative? - Deterministic -> probabilistic - *How much computation* is enough? ::: {.content-visible when-format="html"} ![](https://miro.medium.com/v2/resize:fit:720/format:webp/1*oA63oJ2qoBYiIQ4Wzy8YKw.png){.absolute bottom=50 right=50 width=600px} ::: ## Crash course on statistical mechanics (3) Let's reintroduce the microscopic picture by Boltzmann ![](./public/boltzmann-intro.png){.centered} ## Crash course on statistical mechanics (4) - **Microstate**: a state in the phase space $\{\mathbf{x}_i, \mathbf{p}_i\}$ - **Macrostate**: a specification of macroscopic constraints (e.g. $<E, V, N>$) that corresponds to many microstates - **Ensemble**: a set of constraints like $<E, V, N>$ describing the system ::: {.callout-note} ##### Central principle linking macrostate and microstates: The observed macroscopic thermodynamic property $A_{obs}$ is an **ensemble average** of all possible microstates $i$ $$ A_{obs} = \langle A \rangle = \sum_i A_i \, p_i $$ ::: ## Crash course on statistical mechanics (5) Consider a *canonical ensemble* (constant $N, V, U$) ^[Adapted from Green, Kinetics Transport, and Structure in Hard and Soft Materials, Ch 1] - Inner system (I): $U_I$, $\Omega(U_I)$ - Reservoir system (R): $U_R$, $\Omega(U_R)$ - Total system (universe, T): $U_T = U_I + U_R$, $\Omega(U_T) = \Omega(U_I) \times \Omega(U_R)$ - $U_R \gg U_I$ I and R can only exchange internal energy, not particles or volume. Q: The total internal energy of the universe $U_T$ is fixed. How do the inner and reservoir systems balance their energies? ## Crash course on statistical mechanics (6) Let's first see the equilibrium state. - Assume at e.q., we have $U_I = U_I^{0}$ - Probability of inner energy follows how many states in the reservoir they can exchange with: $$P(U_I) \propto \Omega(U_T - U_I^0)$$ - Since $U_T \approx U_R \gg U_I^0$, we can use Taylor series for $\Omega(U_T - U_I^0)$: $$ \ln\!\Omega(U_T - U_I^0) = \ln\!\Omega(U_T) - \frac{\partial \ln\!\Omega}{\partial U} \bigg\vert_{U=U_T} \cdot U_I^0 + \dots $$ ## Crash course on statistical mechanics (7) - We know the Boltzmann equation $S = k_B \ln\!\Omega$ $$ \ln P(U_I^0) \propto \ln\!\Omega(U_T) - \frac{\partial S_T}{k_B \partial U} \bigg\vert_{U=U_T} \cdot U_I^0 $$ Recall the classical thermodynamic relation: $$ dU = TdS - pdV $$ We have $\dfrac{\partial S}{\partial U}\bigg\vert_{V} = \dfrac{1}{T}$, therefore $$ P(U_I^0) \propto \Omega(U_T) \exp(-\dfrac{U_I^0}{k_B T}) $$ This leads the famous **Boltzmann distribution** ## Crash course on statistical mechanics (8) What if we're not at equilibrium? $T_I$ and $T_R$ may be two distinct values. - We will apply *local equilibrium* assumption here - One postulate for stat. mech. is that the entropy of the universe always **increases**. - We go further by say the equilibrium **maximizes** the universe entropy $$ \left( \dfrac{ \partial \ln \Omega(U_T) }{\partial U_I} \right)_{N, V, U} = 0 $$ - *Can we make sure it is the maximum?* - What is the sign of $\partial^2 S/\partial U^2$? ## Crash course on statistical mechanics (9) $$ \left( \dfrac{ \partial \ln \Omega(U_I) }{\partial U_I} \right)_{N, V, U} + \left( \dfrac{ \partial \ln \Omega(U_R) }{\partial U_I} \right)_{N, V, U} = 0 $$ Use the relation $\partial U_I = -\partial U_R$ $$ \left( \dfrac{ \partial \ln \Omega(U_I) }{\partial U_I} \right)_{N, V, U} = \left( \dfrac{ \partial \ln \Omega(U_R) }{\partial U_R} \right)_{N, V, U} $$ Apply Boltzmann equation $$ T_I^{-1} = T_R^{-1} $$ Yay! This is something we know, temperature at equilibrium. ## What have we observed so far? {.center} - Probability distribution **at equilibrium** follows Boltzmann distribution (for canonical ensemble) - Equilibrium is a consequence of maximizing entropy of the universe - We can locally apply the equilibrium thermodynamics to describe kinetic process (though we don't know how fast yet) ## Formal framework In Onsager's Irreversible Thermodynamics, for any thermodynamic parameter $X$, we define - **Affinity** $F$: the influence of $X_i$ to the universe entropy $$ F_i = \frac{\partial S}{\partial X_i} $$ - **Flux** $J$: the change of parameter $X_i$ (or density of it) per unit time $$ J_i = \frac{d X_i}{d t} $$ - **Kinetic Coefficient** $L_{ij}$: influence of flux type $i$ by affinity type $j$ (linear term). $$ L_{ij} = \dfrac{\partial J_i}{\partial F_j} $$ ## Postulate 1: local equilibrium - It is generally possible to use principles of the **continuum limit** to define meaningful, useful, local values of various thermodynamics quantities, e.g., chemical potential $\mu_i$. - Useful kinetic theories can be developed by assuming a functional relationship between the rate of a process and the local departure from equilibrium (“**driving force**”). ## Postulate 2: entropy production Irreversible Process is An Increase of Entropy in the Universe $$ \frac{dS}{dt} = \sum_i \frac{\partial S}{\partial X_i} \frac{\partial X_i}{\partial t} = \sum_i F_i J_i $$ - Entropy can be created in the universe - Entropy can flow between micro-systems - Rate of entropy production (entropy generation + flow) must be positive ## Postulate 3: linearity - The relation between $J$ and $X$ can be linearly described (resistive system). - Flux is only dependent on the affinity **at the same time** ```{=tex} \begin{align} J_{1} &= L_{11} F_{1} + L_{12}F{2} + \cdots + L_{1\Sigma}F_{\Sigma}\\ \vdots &\\ J_{\Sigma} &= L_{\Sigma1} F_{1} + L_{\Sigma2}F{2} + \cdots + L_{\Sigma\Sigma}F_{\Sigma}\\ \end{align} ``` Or in vector term $$ \mathbf{J} = \mathbf{L}\cdot\mathbf{F} $$ - Coefficient $L_{ii}$ is called **direct kinetic coefficient** - Affinity $F_{i}$ is called **conjugate driving force** w.r.t. $J_i$ - Coefficient $L_{ij}(i\neq j)$ is called **coupling coefficient** ## Postulate 4: reciprocity Onsager proposed that $$ L_{ij} = L_{ji} $$ This is a key component in material kinetics! ![](https://www.ntnu.edu/documents/1412485/1277516986/Onsager_heading.jpg){width=800} ## Summary - Irreversible processes are driven by **thermodynamic forces** - Each force generates a corresponding **flux** - Near equilibrium, fluxes depend **linearly** on forces - Empirical transport laws (Fourier, Ohm, Fick) are special cases of: $$ J_i = \sum_j L_{ij} F_j $$ - Onsager reciprocity enforces symmetry: $$ L_{ij} = L_{ji} $$