The Laws of Thermodynamics for Diffusion in Graphs

1. Introduction

Thermodynamics provides a universal framework for understanding the evolution of physical systems. Since the pioneering work of Clausius in the nineteenth century [1], the First and Second Laws of Thermodynamics have been recognized as fundamental principles governing energy conservation and entropy production. Boltzmann later established the probabilistic foundation of entropy [2], and Shannon extended the concept to information theory [3]. These laws remain central not only to physics but also to many areas of science and engineering, where mathematical models must ensure that system evolution is described in a physically consistent way.

The fundamental principles can be stated in broad terms as follows:

• First Law of Thermodynamics (Conservation of Energy). The total energy of an isolated system must remain constant during its time evolution. Energy may change form, but it cannot be created or destroyed.
• Second Law of Thermodynamics (Non-decrease of Entropy). The entropy of an isolated system can never decrease during its time evolution. Entropy may remain constant in reversible processes, but it increases in irreversible processes.

These laws provide the conceptual framework for assessing the consistency of mathematical models, namely their ability to faithfully represent the physically consistent evolution of a system. This perspective is relevant not only to physics but also to many areas of science and engineering.

In the following sections, we verify that these principles hold in two different settings related to diffusion processes such as heat conduction: first in the continuum setting, and secondly in the discrete setting of weighted undirected graphs. The validity of thermodynamic laws in the continuum setting is a classical and well-established result, which we present here since it serves as the motivating framework (Section 2). By contrast, the proofs of validity in the graph setting represent the central contribution of the present paper (Section 3). Such discrete formulations are increasingly relevant in applications ranging from network dynamics to data science and biological systems [4,5,6,7], and our results provide a rigorous guarantee of physical consistency for graph diffusion models.

2. On the Satisfaction of the Thermodynamic Laws by the Continuum Diffusion Model

In this section we examine the continuum diffusion model in order to verify its consistency with the fundamental laws of thermodynamics. We present the standard diffusion model in the continuum as if it were a heat conduction problem, with the diffusive variable interpreted as the temperature u. This physical interpretation provides an intuitive framework, but it should be emphasized that the mathematical formulation applies in general to other diffusion phenomena. Our aim is to show that the diffusion equation, expressed in the form of heat conduction, satisfies both the conservation of energy (First Law) and the non–decrease of Clausius entropy (Second Law). To this end, we first introduce the governing partial differential equation together with its boundary and initial conditions, which provide the framework for defining internal energy and entropy in the continuum setting.

2.1. Heat Diffusion Model in the Continuum

In the continuum, the evolution of the absolute temperature $u(x,t)$ in a homogeneous material domain $\mathrm{\Omega }$ with smooth boundary $\partial \mathrm{\Omega }$ is modeled by the diffusion equation

$${\displaystyle \frac{\partial u}{\partial t}}(x,t)=\mathrm{\Delta }u(x,t),x\in \mathrm{\Omega },t>0,$$

(1)

where $u(x,t)$ denotes the absolute temperature field, and $\mathrm{\Delta }u$ denotes the Laplacian operator, which is equal to the divergence of the gradient of u:

$$\mathrm{\Delta }u=\nabla ·(\nabla u)$$

(2)

For simplicity, we assume that the material density is $\rho =1$, the thermal conductivity is $k=1$, and the specific heat $c_v$ is constant.

Remark 1.

In the continuum setting, we refer to the operator Δ as the Laplacian operator. This corresponds to the classical Laplace operator in partial differential equations. The use of the term “Laplacian” ensures consistency with the discrete case, where the analogous operator is the Graph Laplacian.

The diffusion equation (1) is completed with proper boundary conditions and initial conditions. Regarding boundary conditions, we will assume here that the domain is fully isolated from the exterior, so:

$${\displaystyle \frac{\partial u}{\partial \overrightarrow{n}}}(x,t)=0,x\in \partial \mathrm{\Omega },t>0,$$

(3)

Here ${\displaystyle \frac{\partial u}{\partial \overrightarrow{n}}}$ is the normal derivative of u at each point of the boundary $\partial \mathrm{\Omega }$. The boundary condition ${\displaystyle \frac{\partial u}{\partial \overrightarrow{n}}}=0$ enforces that the material is thermally isolated from its surroundings. This boundary condition allows us to assess the conservation properties of the model without external influence. Finally, since we want to prove conservative properties for all solutions u, we choose an arbitrary positive initial temperature distribution $u^0\left(x\right)>0$ as initial condition, that is to say:

$$u(x,0)=u^0\left(x\right),x\in \mathrm{\Omega },$$

(4)

Note that a measure of thermal disequilibrium is given by the so–called Dirichlet energy:

$$\mathcal{D}\left(u\right)={\displaystyle \frac{1}{2}}{\int }_{\mathrm{\Omega }}{|\nabla u(x,t)|}^{2}dx.$$

(5)

From this expression, it can be shown that the Euler–Lagrange operator $E\left(\mathcal{D}\right(u\left)\right)$ of such energy is

$$E\left(\mathcal{D}\right(u\left)\right)=-\mathrm{\Delta }u.$$

(6)

Remark 2.

Note from Eq. (6) that, in the continuum, the Laplacian operator $\mathrm{\Delta }u$ is precisely minus the Euler–Lagrange operator of the Dirichlet energy $\mathcal{D}\left(u\right)$. This fundamental variational property will be recalled and used later in the discrete graph setting, where an analogous relation holds between the graph Laplacian and the corresponding Dirichlet energy on graphs.

Remark 3.

Although the present continuum model, defined through Eqs. (1)–(4), arises in the context of heat conduction, the same equations apply to other diffusion phenomena, making the model a general framework for consistency with thermodynamic laws.

In the next subsections, we show that the First and Second Laws of Thermodynamics, stated in the Introduction, are satisfied by the mathematical model given by the heat diffusion equations (1)–(4). In particular, we prove that the total energy remains constant in time, and that the total Clausius entropy of the system does not decrease, thereby confirming that the continuum formulation is consistent with the fundamental principles of thermodynamics.

2.2. The Continuum Diffusion Model Satisfies the First and Second Laws of Thermodynamics

In this section, we prove that the heat diffusion model (1)-(4) satisfies the First Law of Thermodynamics (conservation of total energy) and second Law of Thermodynamics (non-decrease of total entropy).

To proceed with this verification, note from standard classical thermodynamics (see for example, [8]) that the specific thermal energy e in the material is simply proportional to the temperature u, so,

$$e=c_{v}u$$

(7)

From this it follows that the total energy in $\mathrm{\Omega }$ at any time t is given by

$$\mathcal{E}\left(u\right)\left(t\right)={\int }_{\mathrm{\Omega }}e(x,t)d\mathrm{\Omega }=c_v{\int }_{\mathrm{\Omega }}u(x,t)d\mathrm{\Omega }.$$

(8)

Similarly, for this conduction model, Clausius’s specific entropy is given by

$$s=c_{v}log\left(u\right)$$

(9)

where log denotes the natural logarithm. From this it follows that the total entropy in $\mathrm{\Omega }$ at each time t is given by

$$\mathcal{S}\left(u\right)\left(t\right)={\int }_{\mathrm{\Omega }}s(x,t)d\mathrm{\Omega }=c_v{\int }_{\mathrm{\Omega }}logu(x,t)d\mathrm{\Omega }.$$

(10)

Having defined the total energy and the total Clausius entropy in the above formulas at any time t, we can proceed to the proofs of the the laws, as shown in the following two propositions.

Proposition 1

(The Continuum Diffusion Model satisfies the First Law of Thermodynamics). The total energy $\mathcal{E}\left(u\right)\left(t\right)$ of any solution u of the standard diffusion model given by Eqs. (1)-(3) is conserved along time t, that is

$${\displaystyle \frac{d\mathcal{E}\left(u\right)}{dt}}\left(t\right)=0\mathrm{for}\mathrm{all}t.$$

Proof. 

Differentiating Eq. (8) with respect to time t, and using the model diffusion equation (1) under the integral sign, we get

$${\displaystyle \frac{d\mathcal{E}\left(u\right)}{dt}}\left(t\right)=c_v{\int }_{\mathrm{\Omega }}{\displaystyle \frac{\partial u}{\partial t}}(x,t)d\mathrm{\Omega }=c_v{\int }_{\mathrm{\Omega }}\mathrm{\Delta }u(x,t)d\mathrm{\Omega }.$$

By Gauss’ Divergence Theorem,

$${\int }_{\mathrm{\Omega }}\mathrm{\Delta }u(x,t)d\mathrm{\Omega }={\int }_{\mathrm{\Omega }}\nabla ·(\nabla u(x,t))d\mathrm{\Omega }={\int }_{\partial \mathrm{\Omega }}{\displaystyle \frac{\partial u}{\partial \overrightarrow{n}}}(x,t)d\sigma =0,$$

since the isolation from the exterior boundary condition (3) is ${\displaystyle \frac{\partial u}{\partial \overrightarrow{n}}}=0$. Then, it follows that:

$${\displaystyle \frac{d\mathcal{E}\left(u\right)}{dt}}\left(t\right)=0$$

Hence energy is conserved. □

Proposition 2

(The Continuum Diffusion Model satisfies the Second Law of Thermodynamics). The total entropy $\mathcal{S}\left(u\right)\left(t\right)$ of any solution u of the diffusion model given by Eqs. (1)–(4) never decreases over time t, that is

$${\displaystyle \frac{d\mathcal{S}\left(u\right)}{dt}}\left(t\right)\ge 0\mathrm{for}\mathrm{all}t.$$

Proof. 

Differentiating Eq. (10) with respect to time t, and using the diffusion equation (1) under the integral sign, we obtain

$${\displaystyle \frac{d\mathcal{S}\left(u\right)}{dt}}\left(t\right)=c_v{\int }_{\mathrm{\Omega }}{\displaystyle \frac{1}{u(x,t)}}{\displaystyle \frac{\partial u}{\partial t}}(x,t)d\mathrm{\Omega }=c_v{\int }_{\mathrm{\Omega }}{\displaystyle \frac{1}{u(x,t)}}\mathrm{\Delta }u(x,t)d\mathrm{\Omega }.$$

Integrating by parts (Green’s first identity) and applying the boundary condition,

$${\int }_{\mathrm{\Omega }}{\displaystyle \frac{1}{u}}\mathrm{\Delta }ud\mathrm{\Omega }=-{\int }_{\mathrm{\Omega }}\nabla \left({\displaystyle \frac{1}{u}}\right)·\nabla ud\mathrm{\Omega }={\int }_{\mathrm{\Omega }}{\displaystyle \frac{{|\nabla u|}^2}{u^2}}d\mathrm{\Omega }\ge 0.$$

Therefore, ${\displaystyle \frac{d\mathcal{S}\left(u\right)}{dt}}\left(t\right)\ge 0$, showing that the total entropy of the system never decreases. □

In summary of the above two propositions, we can state the following corollary:

Corollary 1.

The heat or diffusion equation on continuum satisfies the fundamental thermodynamic principles of 1) conservation of energy and 2) non–decrease of Clausius entropy.

2.3. Non-Decrease of Shannon Entropy in the Continuum Diffusion Model

In this subsection, we show that not only the total Clausius entropy $\mathcal{S}$ satisfies the property of non-decrease over time in the continuum diffusion model, but that the so-called Shannon entropy $\mathcal{H}$ also satisfies this property. The latter is a function of the temperature field u, and it is defined as follows:

$$\mathcal{H}\left(u\right)\left(t\right)=-{\int }_{\mathrm{\Omega }}u(x,t)logu(x,t)dx.$$

(11)

Proposition 3

(Non–decrease of Shannon Entropy in the Continuum Diffusion Model). The Shannon entropy $\mathcal{H}\left(u\right)\left(t\right)$ of any positive solution u of the diffusion model given by Eqs. (1)-(4) never decreases over time t, that is

$${\displaystyle \frac{d\mathcal{H}\left(u\right)}{dt}}\left(t\right)\ge 0\mathrm{for}\mathrm{all}t.$$

Proof. 

Differentiating (11) with respect to t gives

$${\displaystyle \frac{d\mathcal{H}\left(u\right)}{dt}}\left(t\right)=-{\int }_{\mathrm{\Omega }}\left({\displaystyle \frac{\partial u}{\partial t}}(x,t)logu(x,t)+{\displaystyle \frac{\partial u}{\partial t}}(x,t)\right)dx.$$

Using the diffusion equation (1) and the conservation of energy, the second term vanishes. Thus,

$${\displaystyle \frac{d\mathcal{H}\left(u\right)}{dt}}\left(t\right)=-{\int }_{\mathrm{\Omega }}\left(\mathrm{\Delta }u\right)(x,t)logu(x,t)dx.$$

Using Green’s First Identity on $\mathrm{\Omega }$:

$${\int }_{\mathrm{\Omega }}v\mathrm{\Delta }udx=-{\int }_{\mathrm{\Omega }}\nabla v·\nabla udx+{\int }_{\partial \mathrm{\Omega }}v{\displaystyle \frac{\partial u}{\partial \overrightarrow{n}}}d\sigma ,$$

(12)

and applying the Neumann boundary condition yields

$${\displaystyle \frac{d\mathcal{H}\left(u\right)}{dt}}\left(t\right)={\int }_{\mathrm{\Omega }}{\displaystyle \frac{{|\nabla u(x,t)|}^2}{u(x,t)}}dx.$$

Since $u(x,t)>0$, the integrand is nonnegative, and therefore

$${\displaystyle \frac{d\mathcal{H}\left(u\right)}{dt}}\left(t\right)\ge 0.$$

□

3. On the Satisfaction of the Thermodynamic Laws by the Graph Diffusion Model

The aim of this section is to demonstrate that the fundamental principles of thermodynamics, namely the conservation of energy (First Law) and the non–decrease of entropy (Second Law), remain valid when the diffusion process is modeled on finite graphs. In particular, we show that the standard graph diffusion equation preserves total graph energy and ensures that Clausius graph entropy does not decrease over time. We present the standard graph diffusion model as if it were a heat conduction problem, with the diffusive variable interpreted as the temperature u. This physical interpretation provides an intuitive framework, but it should be emphasized that the mathematical formulation applies in general to other diffusion phenomena on discrete structures.

We begin this section by presenting the necessary mathematical background on undirected weighted graphs, followed by the formulation of the heat diffusion equation in the graph setting.

3.1. Undirected Graphs Definitions

A weighted undirected graph is a couple $\mathcal{G}=(V,W)$, where $V=\{1,\dots ,n\}$ is the set of vertices and W is a nonnegative function on $P_2\left(V\right)=\{\{i,j\}:i,j\in V,i\ne j\}$. We define the set of edges of $\mathcal{G}$ as $E=\{\{i,j\}\in P_2\left(V\right):W\left(\{i,j\}\right)>0\}$. The symmetric matrix $w_{ij}=W\left(\{i,j\}\right)$, with $w_{ii}=0$ for every $i\in V$, gives the weights of the edges. See, for example, [9,10] for classical references, and [11,12] for modern applications.

3.2. Heat Diffusion Model in Graphs

Consider a given weighted undirected graph $\mathcal{G}=(V,W)$. We define the temperature function on the graph as

$$u:V\times [0,\infty )\to \mathbb{R},(i,t)\mapsto u(i,t),$$

So $u(i,t)$ denotes the temperature at vertex $i\in V$ at time t. Similarly to the continuum setting (see Eq. (1)), we can define a diffusion equation of the temperature field in the graph of the form

$${\displaystyle \frac{\partial u}{\partial t}}(i,t)=\left({\mathrm{\Delta }}_{\mathcal{G}}u\right)(i,t)i\in V;t\ge 0$$

(13)

where ${\mathrm{\Delta }}_{\mathcal{G}}$ is the Graph Laplacian operator in the graph setting, which is given by:

$$\left({\mathrm{\Delta }}_{\mathcal{G}}u\right)(i,t)={\displaystyle \sum _{j=1}^n}{w}_{ij}(u(j,t)-u(i,t))$$

(14)

as introduced in [9] and widely used in spectral graph theory and diffusion processes [13].

The heat diffusion equation (13) doesn’t need boundary conditions since we are considering here an isolated graph whose nodes are not connected to the exterior. However, like in the continuum case, we do need an initial condition which is to provide an arbitrary positive initial temperature distribution $u^0\left(i\right)>0$ at $t=0$, so

$$u(i,0)=u^0\left(i\right),i\in V,$$

(15)

To ease the notation, and following the standard notation in graph theory, we will write $u(i,t)$ in subindex form as $u_i\left(t\right)$. Then, with this notation, we can write the graph diffusion equation (13) as

$${\displaystyle \frac{d{u}_i}{dt}}\left(t\right)={\left({\mathrm{\Delta }}_{\mathcal{G}}u\right)}_i\left(t\right)i\in V;t\ge 0$$

(16)

and the corresponding Laplacian operator (14) simply as:

$${\left({\mathrm{\Delta }}_{\mathcal{G}}u\right)}_i\left(t\right)={\displaystyle \sum _{j=1}^n}{w}_{ij}(u_j\left(t\right)-u_i\left(t\right))$$

(17)

with the initial condition (15):

$$u_i\left(0\right)=u_i^0,i\in V,$$

(18)

Remark 4.

There are several reasons to consider ${\mathrm{\Delta }}_{\mathcal{G}}u={\sum }_{j=1}^n{w}_{ij}(u_j-u_i)$ as a Laplace-type operator, perhaps the most intrinsic is given by the fact that ${\mathrm{\Delta }}_{\mathcal{G}}$ is the Euler-Lagrange operator associated to the quadratic Dirichlet energy ${\mathcal{D}}_{\mathcal{G}}\left(u\right)$ in the graph, which is given by,

$${\mathcal{D}}_{\mathcal{G}}\left(u\right)={\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{w}_{ij}{(u_j-u_i)}^2.$$

This is the same property that the continuum Laplacian operator $\mathrm{\Delta }u$ in Eq. (2) fulfills with the corresponding Dirichlet energy $\mathcal{D}\left(u\right)$ in Eq. (5). See Remark 2.

Remark 5.

Note that many texts define the graph Laplacian operator with a minus sign difference with respect to the expression (17). We prefer to use expression (17) so the diffusion equation in graphs exactly resembles the one of continuum and also because of the intrinsic property of the Laplacian operator described in Remark 4.

Although, it will not be used in the demonstration of thermodynamic consistency, note that from the spectral properties of ${\mathrm{\Delta }}_{\mathcal{G}}$, the general solution of the initial value heat problem

$$\left\{\begin{array}{cc}{\displaystyle \frac{\partial u_i}{\partial t}}\left(t\right)={\left({\mathrm{\Delta }}_{\mathcal{G}}u\right)}_i\left(t\right)\hfill & i=1,\dots ,n;t>0\hfill \\ u_i\left(0\right)=u_i^0\hfill & \hfill \end{array}\right.$$

is given by

$$u_i\left(t\right)={\displaystyle \sum _{j=1}^n}{e}^{t{\lambda }_j}\langle u^0,{\psi }^j\rangle {\psi }_i^j,$$

where ${\lambda }_j$ are the nonpositive eigenvalues of ${\mathrm{\Delta }}_{\mathcal{G}}$ and ${\psi }^j$ are the corresponding eigenfunctions with $j=1,\dots ,n$.

3.3. The Graph Diffusion Model Satisfies the First and Second Laws of Thermodynamics

Consider a weighted undirected graph $\mathcal{G}=(V,W)$, and a given positive temperature distribution $u_i\left(t\right)$ at each node i in V. The temperature is a direct measure of the energy, so like in the continuum (see Eq. (7)), the specific energy $e_i$ is given by $e_i=c_v{u}_i$ where $c_v$ is a constant specific heat. Then, the total thermal energy ${\mathcal{E}}_{\mathcal{G}}\left(u\right)$ associated to u at time t, is the sum of nodal energies $e_i$ in the graph:

$${\mathcal{E}}_{\mathcal{G}}\left(u\right)\left(t\right)={\displaystyle \sum _{i=1}^n}{e}_i\left(t\right)=c_v{\displaystyle \sum _{i=1}^n}{u}_i\left(t\right).$$

(19)

Likewise, we use the expression of specific Clausius entropy given in Eq. (9), to determine the specific entropy at each node i of the graph as $s_i=c_{v}log\left(u_i\right)$. Then, the total thermal entropy ${\mathcal{S}}_{\mathcal{G}}\left(u\right)$ associated to u at time t, is the sum of nodal specific entropies $s_i$ in the graph:

$${\mathcal{S}}_{\mathcal{G}}\left(u\right)\left(t\right)={\displaystyle \sum _{i=1}^n}{s}_i\left(t\right)=c_v{\displaystyle \sum _{i=1}^n}log\left(u_i\left(t\right)\right).$$

(20)

With the above definitions, we can state and prove the main results of this note, which are: 1) conservation of energy (first law of thermodynamics), 2) the non-decrease of the thermal entropy (second law of thermodynamics), as follows.

Proposition 4

(The Graph Diffusion Model Satisfies the First Law of Thermodynamics). Given an arbitrary weighted undirected graph $\mathcal{G}$, the total thermal energy ${\mathcal{E}}_{\mathcal{G}}\left(u\right)\left(t\right)$ of any solution u of the diffusion model given by Eqs. (16)-(17) is conserved over time t, that is to say

$${\displaystyle \frac{d{\mathcal{E}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)=0\mathrm{for}\mathrm{all}t.$$

(21)

Proof. 

Taking time derivatives of the total energy expression given by (19), we have:

$${\displaystyle \frac{d{\mathcal{E}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)=c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{d{u}_i}{dt}}\left(t\right)=c_v{\displaystyle \sum _{i=1}^n}{\left({\mathrm{\Delta }}_{\mathcal{G}}u\right)}_i\left(t\right)$$

(22)

Using the explicit form of the graph Laplacian operator, into the RHS term of the equation above, we have that:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{E}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)& =c_v{\displaystyle \sum _{i=1}^n}\left({\displaystyle \sum _{j=1}^n}{W}_{ij}(u_j\left(t\right)-u_i\left(t\right))\right)\hfill \\ \hfill & =c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}{u}_j\left(t\right)-c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}{u}_i\left(t\right)\hfill \\ \hfill & =c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}{u}_j\left(t\right)-c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}{u}_j\left(t\right)\hfill \\ \hfill & =0,\hfill \end{array}$$

where for the third equality, we have use the symmetry of $w_{i}j$. Then, (21)¸ is proved. □

Proposition 5

(The Graph Diffusion Model Satisfies the Second Law of Thermodynamics). Given an arbitrary weighted undirected graph $\mathcal{G}$, the total thermal entropy ${\mathcal{S}}_{\mathcal{G}}\left(u\right)\left(t\right)$ of any solution u of the diffusion model given by Eq. (16)-(18) never decreases along time t,

$${\displaystyle \frac{d{\mathcal{S}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)\ge 0\mathrm{for}\mathrm{every}t.$$

(23)

Proof. 

$${\displaystyle \frac{d{\mathcal{S}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)=c_v{\displaystyle \frac{d}{dt}}{\displaystyle \sum _{i=1}^n}log{u}_i\left(t\right)=c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{u(i,t)}}{\displaystyle \frac{\partial u_i}{\partial t}}\left(t\right)=c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{u_i\left(t\right)}}{\left({\mathrm{\Delta }}_{\mathcal{G}}u\right)}_i\left(t\right)$$

(24)

Using the explicit expression (17)

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\mathcal{S}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)=c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{\displaystyle \frac{1}{u_i\left(t\right)}}{W}_{ij}\left(u_j\left(t\right)-u_i\left(t\right)\right)\\ \hfill =-c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}+c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}{\displaystyle \frac{u_j\left(t\right)}{u_i\left(t\right)}}.\end{array}$$

Notice that since W is symmetric we have

$$\begin{array}{cc}\hfill {\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}{\displaystyle \frac{u_j\left(t\right)}{u_i\left(t\right)}}& ={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}\left({\displaystyle \frac{u_j\left(t\right)}{u_i\left(t\right)}}+{\displaystyle \frac{u_i\left(t\right)}{u_j\left(t\right)}}\right)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}{\displaystyle \frac{u_j^2\left(t\right)+u_i^2\left(t\right)}{2u_i\left(t\right)u_j\left(t\right)}}.\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{S}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)& =-c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}+c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}{\displaystyle \frac{u_j^2\left(t\right)+u_i^2\left(t\right)}{2u_i\left(t\right)u_j\left(t\right)}}\hfill \\ \hfill & =c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}\left({\displaystyle \frac{u_j^2\left(t\right)+u_i^2\left(t\right)}{2u_i\left(t\right)u_j\left(t\right)}}-1\right)\hfill \\ \hfill & =c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}{\displaystyle \frac{u_j^2\left(t\right)-2u_j\left(t\right)u_i\left(t\right)+u_i^2\left(t\right)}{2u_i\left(t\right)u_j\left(t\right)}}\hfill \\ \hfill & =c_v{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}{\displaystyle \frac{{(u_j\left(t\right)-u_i\left(t\right))}^2}{2u_i\left(t\right)u_j\left(t\right)}}.\hfill \end{array}$$

Since

$${\displaystyle \frac{{(u_j\left(t\right)-u_i\left(t\right))}^2}{2u_i\left(t\right)u_j\left(t\right)}}\ge 0\mathrm{for}\mathrm{every}i,j\in V,$$

we have (23)

$${\displaystyle \frac{d{\mathcal{S}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)\ge 0.$$

□

In summary of the above two propositions, we can state the following corollary:

Corollary 2.

The diffusion equation (16)-(17) on weighted undirected graphs satisfies the fundamental thermodynamic principles of 1) conservation of energy and 2) non–decrease of Clausius entropy.

The Corollary above establishes the thermodynamic consistency of the discrete graph framework, extending the classical continuum theory to the graph setting. The verification of the First and Second Laws in this setting constitutes the main contribution of the present work.

3.4. Non–Decrease of Boltzmann–Shannon Entropy in the Graph Diffusion Model

This section is the graph counterpart of Section 2.3 presented in the continuum diffusion model. Here, we show that the graph total Clausius entropy ${\mathcal{S}}_{\mathcal{G}}$ is not the only entropy satisfying the property of non-decrease over time in the graph diffusion model, but that the so-called Shannon entropy ${\mathcal{H}}_{\mathcal{G}}$ also satisfies this property. In the present graph setting, ${\mathcal{H}}_{\mathcal{G}}$ is defined as follows:

$${\mathcal{H}}_{\mathcal{G}}\left(u\right)\left(t\right)=-{\displaystyle \sum _{i=1}^n}{u}_i\left(t\right)log{u}_i\left(t\right).$$

(25)

Remark 6.

Notice that the Shannon entropy ${\mathcal{H}}_{\mathcal{G}}$ given in Eq. (25) in the graph setting is the natural discrete form of the Shannon entropy presented in the continuum setting $\mathcal{H}$ (see Eq. (11))

Proposition 6

(Non decrease of the Shannon entropy in the Graph Diffusion Model). Given an arbitrary weighted undirected graph $\mathcal{G}$, the total Shannon entropy ${\mathcal{H}}_{\mathcal{G}}\left(u\right)\left(t\right)$ of any solution u of the diffusion model given by Eq. (16)-(18) never decreases along time t,

$${\displaystyle \frac{d{\mathcal{H}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)\ge 0\mathrm{for}\mathrm{all}t.$$

(26)

Proof. 

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{H}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)& =-{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{d}{dt}}\left(u_i\left(t\right)log{u}_i\left(t\right)\right)\hfill \\ \hfill & =-{\displaystyle \sum _{i=1}^n}\left({\displaystyle \frac{d{u}_i}{dt}}\left(t\right)log{u}_i\left(t\right)+u_i\left(t\right){\displaystyle \frac{1}{u_i\left(t\right)}}{\displaystyle \frac{d{u}_i}{dt}}\left(t\right)\right)\hfill \\ \hfill & =-{\displaystyle \sum _{i=1}^n}\left(log{u}_i\left(t\right)+1\right){\displaystyle \frac{d{u}_i\left(t\right)}{dt}}\hfill \\ \hfill & =-{\displaystyle \sum _{i=1}^n}\left(log{u}_i\left(t\right)+1\right){\left({\mathrm{\Delta }}_{G}u\right)}_i\left(t\right)\hfill \\ \hfill & =-{\displaystyle \sum _{i=1}^n}\left(log{u}_i\left(t\right)+1\right){\displaystyle \sum _{j=1}^n}{W}_{ij}(u_j\left(t\right)-u_i\left(t\right)).\hfill \end{array}$$

So,

$${\displaystyle \frac{d{\mathcal{H}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}\left(log{u}_i\left(t\right)+1\right)(u_i\left(t\right)-u_j\left(t\right)).$$

By permutation of indices $j,i$ and symmetry, we can write the right hand side term of the above equation as

$${\displaystyle \frac{d{\mathcal{H}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}\left(log{u}_j\left(t\right)+1\right)(u_j\left(t\right)-u_i\left(t\right)).$$

So, by combining both equations, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathcal{H}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)& ={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}\left[\left(log{u}_j\left(t\right)+1\right)(u_j\left(t\right)-u_i\left(t\right))\right.\hfill \\ \hfill & +\left.\left(log{u}_i\left(t\right)+1\right)(u_i\left(t\right)-u_j\left(t\right))\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}\left[log{u}_j\left(t\right)(u_j\left(t\right)-u_i\left(t\right))\right.\hfill \\ \hfill & +\left.log{u}_i\left(t\right)(u_i\left(t\right)-u_j\left(t\right))\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}\left[u_j\left(t\right)(log{u}_j\left(t\right)-log{u}_i\left(t\right))\right.\hfill \\ \hfill & +\left.u_i\left(t\right)(log{u}_i\left(t\right)-log{u}_j\left(t\right))\right].\hfill \end{array}$$

So that

$${\displaystyle \frac{d{\mathcal{H}}_{\mathcal{G}}\left(u\right)}{dt}}\left(t\right)={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{W}_{ij}\left[u_j\left(t\right)-u_i\left(t\right)\right]log{\displaystyle \frac{u_j\left(t\right)}{u_i\left(t\right)}}.$$

(27)

Note now that if a and b are two arbitrary positive real numbers, the function of the real number $(a-b)log{\displaystyle \frac{a}{b}}\ge 0$ is always nonnegative. Then, using this property, in the right hand side of (27), it follows that ${\displaystyle \frac{d\mathcal{H}\left(u\right)}{dt}}\left(t\right)\ge 0$ as desired. □

4. Conclusions

This paper demonstrates that, for graph diffusion models, discrete formulations of the two fundamental laws of thermodynamics can be established and are strictly satisfied by standard diffusion models defined on weighted undirected graphs. In particular, these models satisfy the conservation of graph energy (First Law of Thermodynamics) and the non-decrease of graph Clausius entropy (Second Law of Thermodynamics).

Furthermore, it has been shown that not only does Clausius entropy satisfy the property of non-decreasing system entropy along the evolution process, but the so-called Shannon entropy also satisfies this property, both in the continuum setting and in the graph setting.

The proofs presented here provide a rigorous bridge between continuous thermodynamic theory and discrete network models, opening the way for applications in complex systems, data science, and autonomous networked dynamics. The results highlight the structural consistency of thermodynamic laws across both continuous and discrete domains, and establish the graph case as a natural and mathematically sound counterpart to the classical theory.