MATE 664 Lecture 03

Driving Forces in Irreversible Thermodynamics

Author

Dr. Tian Tian

Published

January 12, 2026

Note

Slides 👉 Open presentation🗒️
PDF version of course note 👉 Open in pdf
Handwritten notes 👉 Open in pdf

Recap of Lecture 02

Key ideas from last lecture:

Crash course on statistical mechanics
- Microstate in phase space ⇔ macroscopic property with probability
- Measurement of microstate ⟹ entropy ⟹ Boltzmann formula
- Explain Boltzmann distribution
- Explain entropy maximizing principle
Introduction to irreversible thermodynamics
- Postulate 1: local equilibrium assumption
- Postulate 2: rate of entropy production is non-zero
- Postulate 3: linear coupling between flux and forces
- Postulate 4: symmetry between coupling coefficients

Learning Outcomes

After today’s lecture, you will be able to:

Recall the entropy production equation and link to rate
Analyze the formula for entropy production the driving force expression
Derive general equation for driving forces in system
Formulate relationship between reciprocal driving force situations

Entropy Balance Equation

General form: \[ \frac{dS}{dt} = \left(\frac{dS}{dt}\right)_{\text{exchange}} + \left(\frac{dS}{dt}\right)_{\text{production}} \]

Exchange: entropy flow across system boundaries
Production: entropy generated internally
Entropy can flow and can be generated

Entropy Production In Microscopic Cells

When applying local equilibrium, we generally consider entropy density $s = S_i / V_i$ for a certain microscopic cell.

Using Balluffi book notations, entropy balance equation looks like:

\[ \frac{\partial s}{\partial t} = -\nabla\cdot\vec{J}_{s} + \dot{\sigma} \]

$\vec{J}_{s}$: flux of entropy (exchange)
$\dot{\sigma}$: entropy generation rate

Irreversible Thermodynamics

Central law:

\[ \dot{\sigma} = \frac{\partial s}{\partial t} + \nabla\cdot\vec{J}_{s} \geq 0 \]

This is the local form of the second law.
Equilibrium $\dot{\sigma} = 0$
Can we have $\frac{\partial s}{\partial t} = 0$?

Entropy Balance: Solving $\dot{\sigma}$ and $\vec{J}_{s}$

Chain rule again! For any thermodynamic quantity $\xi_{i}$:

\[\begin{align} \frac{\partial s}{\partial t} &= \sum_i \frac{\partial s}{\partial \xi_i} \frac{\partial \xi_i}{\partial t} \\ &= \sum_i \frac{\partial s}{\partial \xi_i} (- \nabla \cdot \vec{J}_{\xi_i}) \end{align}\]

What quantities $\xi_i$ can be used?
How do we solve $\frac{\partial s}{\partial \xi_i}$?

Entropy Flux and Production: Results

After arranging the formula for entropy balance and apply laws in vector operations, we get:

\[\begin{align} \vec{J}_s &= \sum_i \frac{\partial s}{\partial \xi_i} \vec{J}_{\xi_i} \\ \dot{\sigma} &= \sum_i \vec{J}_{\xi_i} \cdot \nabla \left(\frac{\partial s}{\partial \xi_i}\right) \end{align}\]

It seems everything has relation with:
- A flux $\vec{J}_{\xi_i}$
- An coefficient $\frac{\partial s}{\partial \xi_i}$ (Onsager’s affinity)
Non-negative entropy production 👉 $\vec{J}_{\xi_i}$ and gradient (vector) $\nabla \left(\frac{\partial s}{\partial \xi_i}\right)$ have an angle no more than 90 degrees!

What Are The Affinities, Anyway?

Consider generalized 1st law of thermodynamics:

\[\begin{align} d s = \frac{1}{T} du - \sum_i \frac{\psi_i}{T} d\xi_i \end{align}\]

It requires:

$\xi_i$ is an extensive property
$\psi_i$ in an intensitve property often called potential
$\xi_i$ is conjugate to $\psi_i$

What are $\psi$ and $\xi$?

General entropy differential (extended variables): \[ ds = \frac{1}{T}du - \frac{p}{T}dv - \sum_i \frac{\psi_i}{T} d\xi_i \]

$\xi_i$: extensive variables
- volume $v$
- charge $q$
- concentration $c$
- surface area $A$
- dipole moment $\mathbf{p}$
- magnetic moment $\mathbf{b}$

$\psi_i$: conjugate intensive variables
- pressure $p$
- electric potential $\phi$
- chemical potential $\mu$
- surface energy $\gamma$
- external electric field $\mathbf{E}$
- magnetic field $\mathbf{H}$

Each $(\psi_i,\xi_i)$ pair contributes to entropy change.

Entropy Flux and Production In Driving Force Terms

We can rewrite the entropy balance equation using individual fluxes:

\[\begin{align} \vec{J}_s &= - \sum_i \frac{\psi_i}{T}\,\vec{J}_{\xi_i} = \frac{1}{T}\,\vec{J}_u - \sum_i \frac{\psi_i}{T}\,\vec{J}_{\xi_i} \\ \dot{\sigma} &= - \sum_i \vec{J}_{\xi_i}\cdot \nabla\!\left(\frac{\psi_i}{T}\right) = \vec{J}_u \cdot \nabla\!\left(\frac{1}{T}\right) - \sum_i \vec{J}_{\xi_i}\cdot \nabla\!\left(\frac{\psi_i}{T}\right) \end{align}\]

If $u$ also have an corresponding potential, what will it be?

Entropy Production: With Flux of Heat Term

Often we wanted to write entropy generation using flux of heat $\vec{J}_{Q} = T \vec{J}_s$:

\[\begin{align} T\dot{\sigma} = - \frac{\vec{J}_Q}{T}\cdot \nabla T - \sum_i \vec{J}_{\xi_i}\cdot \nabla \psi_i \end{align}\]

Why is there a negative sign before each term?
Can you generalize this term?

General Formula for Entropy Production

\[\begin{align} \boxed{ \dot{\sigma} = - \sum_i \left[\text{flux of } \xi_i\right] \cdot \left[\nabla(\text{conjugate potential } \psi_i)\right] } \end{align}\]

$\xi_i$ must be extensive $\Rightarrow$ scales with system size $V_i$
$\psi_i$ is the conjugate intensive potential
- does not scale with $V_i$
$\nabla \psi_i$ acts as the thermodynamic driving force

This compact form unifies all irreversible transport processes.

What Are The Driving Forces (From Irreversible Thermodynamics)?

Some quantities and their driving forces, see Balluffi Table 2.1

Extensive quantity $\xi_i$	Flux $\vec{J}_{\xi_i}$	Conjugate force	Potential $\psi_i$
Heat $Q$	$\vec{J}_Q$	$-\frac{1}{T}\nabla\!T$	$\dfrac{1}{T}$
Chemical component $i$	$\vec{J}_i$	$-\nabla \mu_i$	$\mu_i$
Electric charge $q$	$\vec{J}_q$	$-\nabla \phi$	$\phi$

Each row represents one irreversible transport process
Force $=$ gradient of the conjugate intensive potential
Flux $\cdot$ Force contributes to entropy production

–

Empirical Laws of Fluxes

We also know that fluxes are having empirical relation to the driving force (in a gradient form):

\[ \text{Flux} = (\text{transport coefficient}) \times (\text{driving force}) \]

Fourier’s law (heat conduction) \[ \vec{J}_Q = -k \nabla T = -k T^2 \nabla\!\left(\frac{1}{T}\right) \]

Fick’s law (mass diffusion) \[ \vec{J}_i = -D_i \nabla c_i = -\mu_i c_i \nabla \mu_i \]

Ohm’s law (electrical conduction) \[ \vec{J}_q = -\rho \nabla \phi \]

Entropy Production in the Uncoupled Case

If each flux is influenced by only one dominant driving force (D.F.):

\[ \text{entropy production} = (\text{flux}) \cdot (\text{driving force}) \]

Using linear response: \[ \vec{J} = (\text{coefficient}) \times (\text{D.F.}) \]

we obtain \[ \boxed{ T \dot{\sigma} = (\text{coefficient}) \, \lVert \text{D.F.} \rVert^2 } \]

Consequences of Irreversible Dynamics (Uncoupled Driving Force)

Empirically, all transport coefficients are positive from irreversible thermodynamics
Therefore: \[ \sigma \ge 0 \]
Entropy production vanishes only when: \[ \vec{\sigma} = 0 \quad \Longleftrightarrow \quad \lVert \text{D.F.} \rVert = 0 \]
The second law constrains transport coefficients
Positivity of entropy production $\Rightarrow$ positivity of coefficients
This reasoning will be generalized when fluxes are coupled

Linear Irreversible Thermodynamics: Coupled Fluxes

Extensive quantities: $\xi_i$
Conjugate driving forces: \[ \vec{F}_i \equiv - \nabla \psi_i \]
Associated fluxes: $\vec{J}_i$

We write them in a matrix form as shown in Lecture 2

\[ \vec{J} = \mathbf{L}\,\vec{F} \]

or, component-wise, \[ \vec{J}_i = \sum_j L_{ij}\,\vec{F}_j \]

Entropy Production with Coupling

\[ \dot{sigma} = \sum_i \vec{J}_i \cdot \vec{F}_i = \sum_{i,j} L_{ij}\,\vec{F}_i \cdot \vec{F}_j \]

Second law requires: \[ \dot{sigma} \ge 0 \]
$\mathbf{L}$ must be positive semi-definite

Onsager Reciprocity (Again!)

Under microscopic reversibility: \[ \boxed{ L_{ij} = L_{ji} } \]

Symmetry of $\mathbf{L}$ reduces number of independent coefficients
Fundamental constraint linking kinetics and thermodynamics

Physical Examples of Coupling

Thermoelectric effects (Seebeck / Peltier)
Thermodiffusion (Soret / Dufour)
Electrokinetic transport

Coupling: Mass ↔︎ Thermal – Soret / Dufour Effects

Coupled flux–force relations

Mass flux driven by temperature gradient (Soret effect)
Heat flux driven by concentration gradient (Dufour effect)

Derivation of Sorret thermal conductivity please see lecture handwritten notes

Coupling: Charge ↔︎ Thermal – Seebeck / Peltier Effects

Coupled flux–force relations - Electric current driven by temperature gradient (Seebeck effect) - Heat flux driven by electric field (Peltier effect)

Coupling: Charge ↔︎ Mass – Electrokinetic Phenomena

Coupled flux–force relations - Charge and mass transport are interdependent in ionic systems

General Matrix for Coupling Effects

Example coupling effect matrix given in J. Phase Equilib. Diffus. 2022 43, 640

Some Additional Consequence of Onsager Reciprocity

There are some consequences of Onsager’s reciprocity. As Curie’s symmetry principle (1894, Pierre Curie, not Marie) stated, the cause of an effect needs to follow tensor rank selection rule, that:

Flux – Driving Force pair can only be induced when they have the same tensor rank, or differ by 2.

Tensor rank coupling from Griskey Book 2002

The coupling of momentum-transfer (fluid, 2nd-order) and chemical reaction (0th-order) has only been recently proposed! See J. Chem. Phys. 2022, 157, 084901.

Summary

Entropy balance equation 👉 solutions for flux of entropy and entropy production
Entropy production rate 👉 flux of quantity ✖️ driving force
Flux of quantity 👉 Coefficient ✖️ Driving Force
Examples of reciprocal relation in transport phenomena
Brief introduction of diffusion

--- title: "MATE 664 Lecture 03" subtitle: "Driving Forces in Irreversible Thermodynamics" author: "Dr. Tian Tian" date: "2026-01-12" format: html: {} revealjs: output-file: slides.html pdf: output-file: L03.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L03.pdf) - Handwritten notes 👉 [Open in pdf](./public/L03-notes_annotated.pdf) ::: ::: ::: {.content-visible when-format="revealjs"} ::: {.callout-note} - **Course Website Updated!** [https://tiangroup-uofa.github.io/mate664-kinetics-of-materials/](https://tiangroup-uofa.github.io/mate664-kinetics-of-materials/) <div style="width: 100%; text-align: center;"> <img src="https://quickchart.io/qr?text=https://tiangroup-uofa.github.io/mate664-kinetics-of-materials//&size=250" alt="QR Code" style="width: 150px; height: 150px; display: block; margin: 0 auto;" /> </div> ::: ::: ## Recap of Lecture 02 {.center} Key ideas from last lecture: - Crash course on statistical mechanics - Microstate in phase space ⇔ macroscopic property with probability - Measurement of microstate ⟹ entropy ⟹ Boltzmann formula - Explain Boltzmann distribution - Explain entropy maximizing principle - Introduction to irreversible thermodynamics - Postulate 1: local equilibrium assumption - Postulate 2: rate of entropy production is non-zero - Postulate 3: linear coupling between flux and forces - Postulate 4: symmetry between coupling coefficients --- ## Learning Outcomes {.center} After today's lecture, you will be able to: - **Recall** the entropy production equation and link to rate - **Analyze** the formula for entropy production the driving force expression - **Derive** general equation for driving forces in system - **Formulate** relationship between reciprocal driving force situations --- ## Entropy Balance Equation General form: $$ \frac{dS}{dt} = \left(\frac{dS}{dt}\right)_{\text{exchange}} + \left(\frac{dS}{dt}\right)_{\text{production}} $$ - Exchange: entropy flow across system boundaries - Production: entropy generated internally - Entropy **can flow** and **can be generated** ## Entropy Production In Microscopic Cells When applying local equilibrium, we generally consider **entropy density** $s = S_i / V_i$ for a certain _microscopic cell_. Using _Balluffi book_ notations, entropy balance equation looks like: $$ \frac{\partial s}{\partial t} = -\nabla\cdot\vec{J}_{s} + \dot{\sigma} $$ - $\vec{J}_{s}$: flux of entropy (exchange) - $\dot{\sigma}$: entropy generation rate ![Scheme of microcell and entropy balance](./public/microcell-entropy.svg){width=80%} --- ## Irreversible Thermodynamics Central law: $$ \dot{\sigma} = \frac{\partial s}{\partial t} + \nabla\cdot\vec{J}_{s} \geq 0 $$ - This is the local form of the second law. - Equilibrium $\dot{\sigma} = 0$ - Can we have $\frac{\partial s}{\partial t} = 0$? --- ## Entropy Balance: Solving $\dot{\sigma}$ and $\vec{J}_{s}$ Chain rule again! For any thermodynamic quantity $\xi_{i}$: ```{=tex} \begin{align} \frac{\partial s}{\partial t} &= \sum_i \frac{\partial s}{\partial \xi_i} \frac{\partial \xi_i}{\partial t} \\ &= \sum_i \frac{\partial s}{\partial \xi_i} (- \nabla \cdot \vec{J}_{\xi_i}) \end{align} ``` - What quantities $\xi_i$ can be used? - How do we solve $\frac{\partial s}{\partial \xi_i}$? --- ## Entropy Flux and Production: Results After arranging the formula for entropy balance and apply laws in vector operations, we get: ```{=tex} \begin{align} \vec{J}_s &= \sum_i \frac{\partial s}{\partial \xi_i} \vec{J}_{\xi_i} \\ \dot{\sigma} &= \sum_i \vec{J}_{\xi_i} \cdot \nabla \left(\frac{\partial s}{\partial \xi_i}\right) \end{align} ``` - It seems everything has relation with: - A flux $\vec{J}_{\xi_i}$ - An coefficient $\frac{\partial s}{\partial \xi_i}$ (Onsager's affinity) - Non-negative entropy production 👉 $\vec{J}_{\xi_i}$ and gradient (vector) $\nabla \left(\frac{\partial s}{\partial \xi_i}\right)$ have an angle no more than 90 degrees! --- ## What Are The Affinities, Anyway? Consider generalized 1st law of thermodynamics: ```{=tex} \begin{align} d s = \frac{1}{T} du - \sum_i \frac{\psi_i}{T} d\xi_i \end{align} ``` It requires: - $\xi_i$ is an **extensive property** - $\psi_i$ in an **intensitve property** often called **potential** - $\xi_i$ is **conjugate** to $\psi_i$ --- ## What are $\psi$ and $\xi$? - General entropy differential (extended variables): $$ ds = \frac{1}{T}du - \frac{p}{T}dv - \sum_i \frac{\psi_i}{T} d\xi_i $$ ::: columns ::: {.column width="50%"} - $\xi_i$: **extensive variables** - volume $v$ - charge $q$ - concentration $c$ - surface area $A$ - dipole moment $\mathbf{p}$ - magnetic moment $\mathbf{b}$ ::: ::: {.column width="50%"} - $\psi_i$: **conjugate intensive variables** - pressure $p$ - electric potential $\phi$ - chemical potential $\mu$ - surface energy $\gamma$ - external electric field $\mathbf{E}$ - magnetic field $\mathbf{H}$ ::: ::: Each $(\psi_i,\xi_i)$ pair contributes to entropy change. --- ## Entropy Flux and Production In Driving Force Terms We can rewrite the entropy balance equation using individual fluxes: ```{=tex} \begin{align} \vec{J}_s &= - \sum_i \frac{\psi_i}{T}\,\vec{J}_{\xi_i} = \frac{1}{T}\,\vec{J}_u - \sum_i \frac{\psi_i}{T}\,\vec{J}_{\xi_i} \\ \dot{\sigma} &= - \sum_i \vec{J}_{\xi_i}\cdot \nabla\!\left(\frac{\psi_i}{T}\right) = \vec{J}_u \cdot \nabla\!\left(\frac{1}{T}\right) - \sum_i \vec{J}_{\xi_i}\cdot \nabla\!\left(\frac{\psi_i}{T}\right) \end{align} ``` - If $u$ also have an corresponding potential, what will it be? --- ## Entropy Production: With Flux of Heat Term Often we wanted to write entropy generation using flux of heat $\vec{J}_{Q} = T \vec{J}_s$: ```{=tex} \begin{align} T\dot{\sigma} = - \frac{\vec{J}_Q}{T}\cdot \nabla T - \sum_i \vec{J}_{\xi_i}\cdot \nabla \psi_i \end{align} ``` - Why is there a negative sign before each term? - Can you generalize this term? --- ## General Formula for Entropy Production ```{=tex} \begin{align} \boxed{ \dot{\sigma} = - \sum_i \left[\text{flux of } \xi_i\right] \cdot \left[\nabla(\text{conjugate potential } \psi_i)\right] } \end{align} ``` - $\xi_i$ must be **extensive** $\Rightarrow$ scales with system size $V_i$ - $\psi_i$ is the **conjugate intensive potential** - does *not* scale with $V_i$ - $\nabla \psi_i$ acts as the **thermodynamic driving force** This compact form unifies all irreversible transport processes. --- ## What Are The Driving Forces (From Irreversible Thermodynamics)? Some quantities and their driving forces, see _Balluffi Table 2.1_ | Extensive quantity $\xi_i$ | Flux $\vec{J}_{\xi_i}$ | Conjugate force | Potential $\psi_i$ | |---|---|---|---| | Heat $Q$ | $\vec{J}_Q$ | $-\frac{1}{T}\nabla\!T$ | $\dfrac{1}{T}$ | | Chemical component $i$ | $\vec{J}_i$ | $-\nabla \mu_i$ | $\mu_i$ | | Electric charge $q$ | $\vec{J}_q$ | $-\nabla \phi$ | $\phi$ | - Each row represents one irreversible transport process - Force $=$ gradient of the conjugate intensive potential - Flux $\cdot$ Force contributes to entropy production -- ## Empirical Laws of Fluxes We also know that fluxes are having empirical relation to the driving force (in a gradient form): $$ \text{Flux} = (\text{transport coefficient}) \times (\text{driving force}) $$ **Fourier’s law (heat conduction)** $$ \vec{J}_Q = -k \nabla T = -k T^2 \nabla\!\left(\frac{1}{T}\right) $$ **Fick’s law (mass diffusion)** $$ \vec{J}_i = -D_i \nabla c_i = -\mu_i c_i \nabla \mu_i $$ **Ohm’s law (electrical conduction)** $$ \vec{J}_q = -\rho \nabla \phi $$ --- ## Entropy Production in the Uncoupled Case If each flux is influenced by only **one dominant driving force (D.F.)**: $$ \text{entropy production} = (\text{flux}) \cdot (\text{driving force}) $$ Using linear response: $$ \vec{J} = (\text{coefficient}) \times (\text{D.F.}) $$ we obtain $$ \boxed{ T \dot{\sigma} = (\text{coefficient}) \, \lVert \text{D.F.} \rVert^2 } $$ --- ## Consequences of Irreversible Dynamics (Uncoupled Driving Force) - Empirically, all transport coefficients are **positive** from irreversible thermodynamics - Therefore: $$ \sigma \ge 0 $$ - Entropy production vanishes only when: $$ \vec{\sigma} = 0 \quad \Longleftrightarrow \quad \lVert \text{D.F.} \rVert = 0 $$ - The second law constrains transport coefficients - Positivity of entropy production $\Rightarrow$ positivity of coefficients - This reasoning will be generalized when **fluxes are coupled** --- ## Linear Irreversible Thermodynamics: Coupled Fluxes - Extensive quantities: $\xi_i$ - Conjugate driving forces: $$ \vec{F}_i \equiv - \nabla \psi_i $$ - Associated fluxes: $\vec{J}_i$ We write them in a matrix form as shown in [Lecture 2](../L02/) $$ \vec{J} = \mathbf{L}\,\vec{F} $$ or, component-wise, $$ \vec{J}_i = \sum_j L_{ij}\,\vec{F}_j $$ --- ## Entropy Production with Coupling $$ \dot{sigma} = \sum_i \vec{J}_i \cdot \vec{F}_i = \sum_{i,j} L_{ij}\,\vec{F}_i \cdot \vec{F}_j $$ - Second law requires: $$ \dot{sigma} \ge 0 $$ - $\mathbf{L}$ must be **positive semi-definite** --- ## Onsager Reciprocity (Again!) Under microscopic reversibility: $$ \boxed{ L_{ij} = L_{ji} } $$ - Symmetry of $\mathbf{L}$ reduces number of independent coefficients - Fundamental constraint linking kinetics and thermodynamics --- ## Physical Examples of Coupling - Thermoelectric effects (Seebeck / Peltier) - Thermodiffusion (Soret / Dufour) - Electrokinetic transport --- ## Coupling: Mass ↔ Thermal -- Soret / Dufour Effects **Coupled flux–force relations** - Mass flux driven by temperature gradient (Soret effect) - Heat flux driven by concentration gradient (Dufour effect) _Derivation of Sorret thermal conductivity please see [lecture handwritten notes](./public/L03-notes_annotated.pdf)_ ![](./public/sorret.png){width="100%"} ## Coupling: Charge ↔ Thermal -- Seebeck / Peltier Effects **Coupled flux–force relations** - Electric current driven by temperature gradient (Seebeck effect) - Heat flux driven by electric field (Peltier effect) ![](./public/thermoelectro.jpg){width="100%"} ## Coupling: Charge ↔ Mass -- Electrokinetic Phenomena **Coupled flux–force relations** - Charge and mass transport are interdependent in ionic systems ![](./public/electrokinetic.png){width="100%"} ## General Matrix for Coupling Effects Example coupling effect matrix given in _J. Phase Equilib. Diffus._ **2022** 43, 640 ![](./public/coupling-matrix.png){width="60%"} ## Some Additional Consequence of Onsager Reciprocity There are some consequences of Onsager's reciprocity. As Curie's symmetry principle (1894, _Pierre_ Curie, _not Marie_) stated, the cause of an effect needs to follow tensor rank selection rule, that: Flux -- Driving Force pair can only be induced when they have the same tensor rank, or differ by 2. ![Tensor rank coupling from Griskey Book 2002](./public/griskey-tensor-order.png){width="60%"} The coupling of momentum-transfer (fluid, 2nd-order) and chemical reaction (0th-order) has only been recently proposed! See _J. Chem. Phys._ **2022**, 157, 084901. ## Summary - Entropy balance equation 👉 solutions for flux of entropy and entropy production - Entropy production rate 👉 flux of quantity ✖️ driving force - Flux of quantity 👉 Coefficient ✖️ Driving Force - Examples of reciprocal relation in transport phenomena - Brief introduction of diffusion

Extensive quantity \(\xi_i\)	Flux \(\vec{J}_{\xi_i}\)	Conjugate force	Potential \(\psi_i\)
Heat \(Q\)	\(\vec{J}_Q\)	\(-\frac{1}{T}\nabla\!T\)	\(\dfrac{1}{T}\)
Chemical component \(i\)	\(\vec{J}_i\)	\(-\nabla \mu_i\)	\(\mu_i\)
Electric charge \(q\)	\(\vec{J}_q\)	\(-\nabla \phi\)	\(\phi\)