**Table of Contents:**

**Chapter 1: Introduction to Network Thermodynamics**

What Is Network Thermodynamics?

Electronic Network Theory

Complex Systems Theory and the Characteristics of Modern Biology.

The Modeling Relation and Rosen's Distinction

The Explicit Recognition of Topological Contributions to System's Behavior

Physical Systems Theory

A Review of Classical Thermodynamics

**Chapter 2: Equilibrium Thermodynamics:Review and
Vocabulary**

What Is "Thermodynamic" Reasoning?

Callen's First Postulate

The Extremum Principle and the Remainder of Callen's Postulates

The Chemical Potential as a Constitutive Rrelation

The Electrochemical Potential

Examples of the Use of the Chemical Potential to Calculate Relations for Constrained Equilibria

*Example 1: The Nernst Potential**Example 2: Osmosis and Introduction of the Partial Molar Volume**Example 3: The Gibbs-Donnan Equilibrium*

The Gibbs-Donnan Equilibrium Obeys the Nernst Equation as an Extremely Good Approximation

An aside: Application to Resting Cells

**Chapter 3: Non-equilibrium Thermodynamics: From Onsager
toPrigogine **

The Nature of Stationary States away from Equilibrium (Steady States)

Entropy Production During an Irreversible Process and the Direction of Heat Flow

The Isothermal Dissipation Function

The Practical Phenomenological Equations

Chemical Reactions, Active Transport and Curie's Principle

**Chapter 4: The Basic Ideas of Network Thermodynamics: I.
The Constitutive Laws and Dynamic Systems **

From Thermodynamics to Dynamic Systems: an Overview

Through and Across Variables

The Building Blocks: Constitutive Relations

Simple Independent Storage Events and the Generalization of Capacitance

Some Independent Inertial Processes

**Chapter 5: The Basic Ideas of Network Thermodynamics: II.
Graph Theoretical Methods for the Encoding of System Topology .**

*A Brief History of Graph Theory*

*The Konigsberg Bridges Problem**Kirchhoff's Theory of Trees (1847)**Chemical Graphs - Cayley (1851)**Hamiltonian Paths**The Four Color Problem*-
*Graph Theory and Its Applications*

Introduction to the Study of Networks and Graphs

*A Short Introduction to Graph Theory**Incidence Matrices**What is a Network?**Network Elements and N-ports*

An Example: The Network Approach to Some Rate Processes

*An Electric Circuit**The Case of Simple Diffusion Across a Membrane**Pressure Driven Volume Flow Through a Membrane (Filtration)**A Simple Version of Chemical Kinetics*

**Chapter 6: Network Thermodynamic Solutions to Steady State
Linear Systems: Nodal and Mesh Analysis and Duality**

The Systematic Approach to Networks

The Intimate Relation Between the Network's Topology and Kirchhoff's Laws

Nodal Analysis of a Network

The Generalized Dissipative Branch

Duality in Networks: Loop-Mesh Analysis is Equivalent to Node Analysis on the Dual

Dual Networks

Loop and Mesh Analysis

**Chapter 7Dynamic Linear Systems and Cutsets: The Network
Thermodynamic Generation of State Vector Equations**

Time Dependent or "Dynamic" Linear Networks

Cut Set Analysis of Time Dependent Linear Networks

Cut Set Analysis: A Method Which Extends to Networks with Nonlinear Resistors, Capacitors and Inductors

*Cut Sets and Fundamental Cut Sets**Cut Set Analysis of Networks**KCL in Fundamental Cut Sets**KVL in Cut Sets*

Generalized Cut Set Analysis and the Analysis of RC Networks

State Vector Equations for Continuous-time Dynamic Networks

Nonlinear Dynamic Systems

The Use of the Proper Tree to Organize the Analysis

Using This Approach on Nonlinear Networks

*Nonlinear Dissipative (Steady State) Networks*

**Chapter 8: N-ports and Tellegen's Theorem: Energy
Conversion, Degree of Coupling, Efficiency and Metrics** . . .
. . . .

Non-equilibrium Thermodynamics Provides the Phenomenological Description of N-ports

Network Thermodynamics Provides a Holistic View of the N-port

The Degree of Coupling and Efficiency of Energy Conversion in N-ports

The Degree of Coupling's Geometric Significance: the Metric and Its Demonstration of the Cannonical Nature of the Network Representation

The Phenomenological Approach

Phenomenology in Hierarchical Systems

Tellegen's Theorem and the Holistic View of Networks

*The Quasi-power Theorem.**Tellegen's Theorem and the Onsager Reciprocal Relations.*-
*Minimum Dissipation and Tellegen's Theorem*

Simplification of Computation in Linear Networks

**Chapter 9: Applications of the N-port Concept: Topological
Contributions to System Behavior, Multiport Current Dividers, and
Other Innovations**

Toward a Biological Circuit Theory

The Current Divider Principle in 1-port Networks

The Node-branch Incidence Matrix for the Current Divider

The Extension of the Current Divider Principle to N-ports

Mathematical Description of a Solute-volume Flow 2-port

**Chapter 10: Making Nonlinearity Look Linear: The Reference
State, Kinetics, and Non-equilibrium thermodynamics Get Married **

The Relationship Between Kinetics and Onsager's Thermodynamics

Onsager's Triangle Reaction as a Network

Nonlinear Systems as Networks and the Reference State

The Need for Reference States: A Simple Kinetic Example

The Reference State

*Non-equilibrium Thermodynamics*

*The Hill Diagram Method*

The Simple Carrier Model: Hill's Method of Analysis

Peusner's "Thermokinetic" Networks

The Creation of Peusner Networks for Thermokinetic Systems

*The Three State Model (General Form)**First Order vs. Pseudo-first Order Transitions*

The Principal of Detailed Balance: First Order vs. Pseudo-first Order Steps in the Network

The Choice of Reference State

An Example with Two Degrees of Freedom: Linear Description and Onsager Coefficients for a Given Reference State

- Solving for the State Populations
- The Thermodynamic Network and the Onsager Coefficients

** Chapter 11: Simulation of Network Models Using SPICE:
Making the Difficult Seem Easy **

SPICE as a General Purpose Simulator

Using Resistors, Capacitors and Constant Sources

*Compartments are Represented by Nodes and Have a "Potential" Assigned to Them**Barriers or Kinetic Elements*

Some Examples

*Sample SPICE Program "Test.cir"**Program Control**Time Constants**The Role of a Volume Capacitance in a Diffusion Network**The Steady State Solution*

Controlled Sources, the "Magic" of SPICE

*First Order Kinetics: Spice Has a Special "Default" Syntax*-
*Second Order Reactions* *Michaelis-Menten Kinetics**Finding Quotients and Reciprocals**Coupled Solute and Solvent Flow*

**Chapter 12: Applications and Models: Compartmental
Systems, Pharmacokinetics, Reaction Networks and Cancer
Chemotherapy**

Applications of Kinetics to Systems

The Compartmental Model of Frog Skin

The Energetics of Coupled Processes and Its Relationship to Compartmental Analysis

A Study of the Energetics of Sodium Transport in Frog Skin

Pharmacokinetic Models

Reaction Networks and Cancer Chemotherapy

Cardiovascular Models

Ecological Models

Structural Identifiability: What Network Thermodynamics Has to Offer

- The First Example: A Linear Three Compartment System Arranged in Series withInput and Output (Sampling) on Opposite Ends

- The Network Solution to the Problem
- Simulation of the System
- Another Example, the Role of Detailed Balance

Appendix: A Short Review of Laplace Transforms

**Chapter 13: Applications and Models: Epithelial Membranes
in Kidney, Gut, and the Lingual Epithelium**

The Epithelial Membrane Revisited

The Model of Coupled Solute/Volume Flow Across Loose Epithelia

The Tight Epithelium Model for Ion Transport and Electrophysiological Responses

*Basic Topology**Constitutive Relations**Physical Constraints**Network Representation**Programming on SPICE2**Parameter Estimation*

**Chapter 14: Epilogue: Towards a Theory of Complex Systems**

Networks, Organization, Complexity and Life

**Appendix: Some Topological Considerations in
Thermodynamics: An Application of Network Thermodynamics **

Onsager Reciprocity as a Topological Property

Reciprocity Defined as a Class of Simultaneous Properties

Reciprocity in Equilibrium Systems and the Existence of Potential Functions

Caratheodory's Proof and the Second Law of Thermodynamics

The Accessibility Theorem of Caratheodory

Proof of the Accessibility Theorem

Forms and the Exterior Calculus

*1-Forms in E3**p-Forms on a Manifold M.*

Reciprocity in Classical Thermodynamics: Maxwell's Relations. . .

The Meaning of Receprocity in a Non-Equilibrium (Dynamic) System

The Topological Implications of Reciprocity

*The De Rham Cohomology*