Thermodynamic and Stoichiometric Laws Ruling the Fates of Growing Systems

We consider the following thermodynamic setup in the present paper (Fig. 1). A growing open chemical reaction system is surrounded by an environment. We assume that the system is always in a well-mixed state (a local equilibrium state), and therefore, we can completely describe it by extensive variables $(E,\Omega,N,X)$. Here, $E$ and $\Omega$ represent the internal energy and the volume of the system. $N=\{N^{m}\}\in\mathbb{R}_{>0}^{\mathcal{N}_{N}}$ denotes the number of chemicals that can move across the membrane between the system and the environment, which we call open chemicals hereafter. $X=\{X^{i}\}\in\mathfrak{X}=\mathbb{R}_{>0}^{\mathcal{N}_{X}}$ is the number of chemicals confined within the system; the indices $m$ and $i$ respectively run from $m=1$ to $\mathcal{N}_{N}$ and from $i=1$ to $\mathcal{N}_{X}$, where $\mathcal{N}_{N}$ and $\mathcal{N}_{X}$ are the number of species of the open and the confined chemicals. The environment is characterized by intensive variables $(\tilde{T},\tilde{\Pi},\tilde{\mu})$, where $\tilde{T}$ and $\tilde{\Pi}$ are the temperature and the pressure; $\tilde{\mu}=\{\tilde{\mu}_{m}\}\in\mathbb{R}^{\mathcal{N}_{N}}$ is the chemical potential corresponding to the open chemicals. Also, $(\tilde{E},\tilde{\Omega},\tilde{N})$ denote the corresponding extensive variables.

In thermodynamics, the entropy function is defined on $(E,\Omega,N,X)$ as a concave and smooth function $\Sigma[E,\Omega,N,X]$. We also write the entropy function for the environment as $\tilde{\Sigma}_{\tilde{T},\tilde{\Pi},\tilde{\mu}}[\tilde{E},\tilde{\Omega},\tilde{N}]$, and the total entropy can be expressed as

$$\Sigma^{\mathrm{tot}}=\Sigma[E,\Omega,N,X]+\tilde{\Sigma}_{\tilde{T},\tilde{\Pi},\tilde{\mu}}[\tilde{E},\tilde{\Omega},\tilde{N}],$$

(1)

where we use the additivity of the entropy. Further, the entropy function for the system has the homogeneity. Therefore, without loss of generality, we can write it as

$$\Sigma[E,\Omega,N,X]=\Omega\sigma[\epsilon,n,x],$$

(2)

where $\sigma[\epsilon,n,x]$ is the entropy density and $(\epsilon,n,x):=(E/\Omega,N/\Omega,X/\Omega)$. In this work, we consider only a situation without phase transition, and therefore, we assume that $\sigma[\epsilon,n,x]$ is strictly concave.

The dynamics of the number of confined chemicals $X(t)$ is described by

$\displaystyle\frac{dX^{i}}{dt}=S^{i}_{r}J^{r}(t),$

(3)

where $J(t)=\{J^{r}(t)\}$ represents the chemical reaction flux; $S=\{S^{i}_{r}\}$ denotes the stoichiometric matrix for the confined chemicals (see Fig. 1). The index $r$ runs from $r=1$ to $\mathcal{N}_{R}$, where $\mathcal{N}_{R}$ is the number of reactions. Also, Einstein’s summation convention has been employed in Eq. (3) for notational simplicity. By integrating Eq. (3), we obtain

$\displaystyle X^{i}(t)=X^{i}_{0}+S^{i}_{r}\Xi^{r}(t),$

(4)

where $X_{0}=\{X^{i}_{0}\}$ denotes the initial condition and $\Xi(t)=\{\Xi^{r}(t)\}$ is the integration of $J(t)$, i.e., the extent of reaction.

In this work, we assume that the stoichiometric matrix $S$ does not have the right kernel space,

$\displaystyle\dim\mathrm{Ker}[S]=0.$

(5)

By contrast, the left kernel space exists and generates a set of conserved quantities:

$\displaystyle L=\{L^{l}=U^{l}_{i}X^{i}_{0}\},$

(6)

i.e., the vector $L\in\mathcal{L}=\mathbb{R}^{\dim\mathrm{Ker}[S^{T}]}$. Here, $U$ is a basis matrix: $\{U_{i}^{l}\}:=(\textbf{U}^{1},\textbf{U}^{2},...)^{T}$ whose row vectors $\textbf{U}^{l}$ form the basis of $\mathrm{Ker}[S^{T}]$, i.e., $US=0$, and $l$ runs from $1$ to $\dim\mathrm{Ker}[S^{T}]$. Since we are mainly interested in the growth of the system, the time evolution of $X(t)$ should perpetually increase the number of all confined chemicals while satisfying the conservation laws (Eq. (6)). In order to realize this situation, the stoichiometric matrix $S$ should satisfy

$$\mathrm{Im}[S]\cap\mathbb{R}_{>0}^{\mathcal{N}_{X}}\neq\emptyset,$$

(7)

which is known as the condition for the productive $S$ [20]. The productive $S$ excludes the mass conservation-type laws, the existence of which trivially precludes the number of confined chemicals to increase continuously.

The number of open chemicals changes through the reactions and the diffusions. By defining the stoichiometric matrix for open chemicals as $O=\{O^{m}_{r}\}$ and the diffusion fluxes as $J_{D}(t)=\{J^{m}_{D}(t)\}$, we have

$\displaystyle\frac{dN^{m}}{dt}=O^{m}_{r}J^{r}(t)+J_{D}^{m}(t).$

(8)

Example 1: To give an illustrative example, we consider the following chemical reactions:

	$\displaystyle R_{1}:$	$\displaystyle A_{1}+A_{3}+B_{1}\rightleftarrows 2A_{2},$
	$\displaystyle R_{2}:$	$\displaystyle A_{2}\rightleftarrows A_{1}+A_{3}+B_{2}.$		(9)

Here, three confined chemicals $A=(A_{1},A_{2},A_{3})$ and two open chemicals $B=(B_{1},B_{2})$ are involved in two reactions $R_{1}$ and $R_{2}$ ¹¹1One may feel that the order of the reactions in Eq. (9) is high. The reason why we consider this example is just because it is straightforward to visualize our geometric representation in Sec. III. Since $\mathcal{N}_{X}=\dim\mathrm{Ker}[S^{T}]+\dim\mathrm{Im}[S]$ and we can visualize up to three dimensional space ($\mathcal{N}_{X}=3$), relevant cases are when $\dim\mathrm{Im}[S]$ is one, two or three. The present example corresponds to the case in which $\dim\mathrm{Im}[S]$ is two. In Sec. I and II of the Supplementary material, we also consider other examples with which $\dim\mathrm{Im}[S]$ is one and three, respectively.. The stoichiometric matrices $S$ and $O$ for the confined and open chemicals, respectively, are

$\displaystyle S=\bordermatrix{&R_{1}&R_{2}\cr A_{1}&-1&1\cr A_{2}&2&-1\cr A_{3}&-1&1\cr},\mbox{ }O=\bordermatrix{&R_{1}&R_{2}\cr B_{1}&-1&0\cr B_{2}&0&1\cr}.$

(10)

Since $\mathcal{N}_{R}=2$ in this case, we can represent the number of confined chemicals by introducing $\Xi=(\Xi^{1},\Xi^{2})^{T}$ as

$$\left(\begin{array}[]{c}X^{1}\\ X^{2}\\ X^{3}\end{array}\right)-\left(\begin{array}[]{c}X_{0}^{1}\\ X_{0}^{2}\\ X_{0}^{3}\end{array}\right)=S\Xi=\left(\begin{array}[]{c}-1\\ 2\\ -1\end{array}\right)\Xi^{1}+\left(\begin{array}[]{c}1\\ -1\\ 1\end{array}\right)\Xi^{2}.$$

(11)

For this example, $\dim\mathrm{Ker}[S^{T}]$ is one, and $U=(-1,0,1)$ satisfies $US=0$. Thus, the reactions have the conserved quantity $L=UX=-X^{1}+X^{3}$. We can intuitively see the conserved quantity from the chemical equations in Eq. (9) as follows. The numbers of $A_{1}$ and $A_{3}$, i.e., $X^{1}$ and $X^{3}$, change in the same manner: each chemical is consumed by one molecule in the forward reaction of $R_{1}$ and is produced by one in that of $R_{2}$. Thus, the difference between $X^{1}$ and $X^{3}$ is conserved when each of the reactions occurs.

$\blacksquare$

In this paper, we assume that the time scale of the chemical reactions is much slower than that of the others, i.e., $J\ll J_{E},J_{\Omega},J_{D}$. This means that we consider the isothermal and isobaric process with a balance of chemical potentials for open chemicals. This assumption allows us to separately analyze the slow dynamics of chemical reactions from the fast dynamics of the energy, the volume, and the chemical diffusion. As shown in Appendix A, our dynamics is effectively governed by Eq. (3), and the other variables $(E,\Omega,N)$ are rapidly relaxed to the values at the equilibrium state of the fast dynamics. As a result, $(E,\Omega,N)=(E(X),\Omega(X),N(X))$ is obtained as a function of $X$.

With the above assumptions, we obtain the following expression for the total entropy (see Eq. (74) in Appendix A),

	$\displaystyle\Sigma^{\mathrm{tot}}(\Xi)=$	$\displaystyle-\frac{1}{\tilde{T}}\left\{\Omega(X)\varphi\left(\frac{X}{\Omega(X)}\right)+\Omega(X)\tilde{\Pi}+\tilde{\mu}_{m}O^{m}_{r}\Xi^{r}\right\}$
		$\displaystyle+\mathrm{const},$		(12)

where $X=X_{0}+S\Xi$. Here, $\varphi(\cdot)$ is the partial grand potential density, which is given by a Legendre transformation of $\sigma\left[\epsilon,n,x\right]$:

$\displaystyle\varphi(x):=\varphi\left[\tilde{T},\tilde{\mu};x\right]=\min_{\epsilon,n}\left\{\epsilon-\tilde{T}\sigma\left[\epsilon,n,x\right]-\tilde{\mu}_{m}n^{m}\right\}.$

(13)

Note that $\varphi(x)$ is strictly convex because $\sigma$ is strictly concave. The volume $\Omega(X)$ is variationally determined under isobaric conditions as

$$\Omega(X)=\arg\min_{\Omega}\left\{\Omega\varphi\left(\frac{X}{\Omega}\right)+\Omega\tilde{\Pi}\right\}.$$

(14)

Example 2: If we assume that the system and the environment are composed of ideal gas or solution, the partial grand potential density $\varphi(x)$ is expressed as

	$\displaystyle\varphi(x)=x^{i}\nu_{i}^{o}\left(\tilde{T}\right)$	$\displaystyle+R\tilde{T}\sum_{i}\left\{x^{i}\log x^{i}-x^{i}\right\}$
		$\displaystyle-R\tilde{T}\sum_{m}e^{\left\{\tilde{\mu}_{m}-\mu_{m}^{o}\left(\tilde{T}\right)\right\}/R\tilde{T}},$		(15)

where $R$ is the gas constant, and $\nu^{o}(\tilde{T})=\{\nu_{i}^{o}(\tilde{T})\}$ and $\mu^{o}(\tilde{T})=\{\mu^{o}_{m}(\tilde{T})\}$ are the standard chemical potentials of the confined and the open chemicals, respectively[78].

Since we assume that the environment also consists of the ideal gas, the chemical potential $\tilde{\mu}$ in Eq. (15) is represented as

$\displaystyle\tilde{\mu}_{m}=\mu_{m}^{o}(\tilde{T})+R\tilde{T}\log\tilde{n}^{m},$

(16)

where $\tilde{n}=\{\tilde{n}^{m}\}$ is the density of the open chemicals in the environment. Furthermore, Eq. (14) gives the following expression for the volume $\Omega$,

$\displaystyle\Omega(X)=\frac{R\tilde{T}\sum_{i}X^{i}}{\tilde{\Pi}-R\tilde{T}\sum_{m}\tilde{n}^{m}},$

(17)

which corresponds to the equation of state.

$\blacksquare$

From Eq. (14), the density $x\in\mathcal{X}=\mathbb{R}_{>0}^{\mathcal{N}_{X}}$ is obtained by a nonlinear function $\rho_{\mathcal{X}}(X)$ of $X$ as

$\displaystyle\rho_{\mathcal{X}}(X):=x(X)=X/\Omega(X)\in\mathcal{X}.$

(18)

Since $\Omega(X)$ is homogeneous, we have $\rho_{\mathcal{X}}(\alpha X)=\rho_{\mathcal{X}}(X)$ for $\alpha>0$. This fact implies that $\rho_{\mathcal{X}}$ induces one-to-one correspondence between rays in the number space $\mathfrak{X}$ and points in the density space $\mathcal{X}$. Here, we write the corresponding ray to a density $x$, i.e., the fiber of $x$ for $\rho_{\mathcal{X}}$, as

	$\displaystyle\mathfrak{r}^{\mathcal{X}}(x):$	$\displaystyle=\Set{X}{\rho_{\mathcal{X}}(X)=x,X>0}\subset\mathfrak{X},$
		$\displaystyle=\Set{X}{X=\alpha x,\alpha>0}.$		(19)

We define the full grand potential density $\varphi^{*}(y)=\varphi^{*}[\tilde{T},\tilde{\mu};y]$ by the Legendre transformation of $\varphi(x)$ in Eq. (13):

$$\varphi^{*}(y):=\max_{x}\left\{y_{i}x^{i}-\varphi(x)\right\}.$$

(20)

We mention that $\varphi^{*}(y)$ is strictly convex. Because of the one-to-one correspondence of the Legendre transformation by $\varphi(x)$ and $\varphi^{*}(y)$, a state of the system is equivalently specified either by the density $x\in\mathcal{X}=\mathbb{R}_{>0}^{\mathcal{N}_{X}}$ of the confined chemicals or by its Legendre dual variable $y=\partial\varphi(x)\in\mathcal{Y}=\mathbb{R}^{\mathcal{N}_{X}}$ ²²2The inverse transformation is given by $x=\partial\varphi^{*}(y)$. The existence of one-to-one correspondence between $\mathbb{R}_{>0}^{\mathcal{N}_{X}}$ and $\mathbb{R}^{\mathcal{N}_{X}}$ is physically reasonable (See Sec. III.1).. The thermodynamic interpretation of $y$ is the corresponding chemical potential to $x$. In addition, $\varphi^{*}(y)$ can be interpreted as the pressure of the system at the state $y$. Recalling the one-to-one correspondence by $\rho_{\mathcal{X}}$ between rays in $\mathfrak{X}$ and points in $\mathcal{X}$, we notice that there exists the one-to-one correspondence between rays in $\mathfrak{X}$ and points in $\mathcal{Y}$. Here, we write the corresponding ray to a chemical potential $y$ as

$\displaystyle\mathfrak{r}^{\mathcal{Y}}(y):=\Set{X}{\partial\varphi\circ\rho_{\mathcal{X}}(X)=y,X>0}\subset\mathfrak{X}.$

(21)

The chemical equilibrium states are given by the balance of chemical potentials between reactants and products [78]:

$$y_{i}S^{i}_{r}+\tilde{\mu}_{m}O^{m}_{r}=0.$$

(22)

Thus, we can obtain the candidates of chemical equilibrium states by the solutions to Eq. (22):

$$\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu}):=\Set{y}{y_{i}S^{i}_{r}+\tilde{\mu}_{m}O^{m}_{r}=0},$$

(23)

which we term the equilibrium manifold. Here, we note that $\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})\neq\emptyset$, because we have assumed $\dim\mathrm{Ker}[S]=0$ in Eq. (5), i.e., $\dim\mathrm{Im}[S^{T}]=\mathcal{N}_{R}$.

The reaction dynamics in Eq. (3) with its conserved quantities $L$ restricts the accessible region in $\mathfrak{X}$ as

$$\mathcal{M}^{\mathfrak{X}}_{\mathrm{STO}}(L):=\Set{X}{U^{l}_{i}X^{i}=L^{l},X>0}.$$

(24)

This subspace is known as the stoichiometric compatibility class[80]. By using the mappings $\rho_{\mathcal{X}}$ in Eq. (18) and $\partial\varphi$, the accessible regions in the density space $\mathcal{X}$ and the chemical potential space $\mathcal{Y}$ are respectively obtained as

	$\displaystyle\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L)$	$\displaystyle=\rho_{\mathcal{X}}(\mathcal{M}^{\mathfrak{X}}_{\mathrm{STO}}(L)),$		(25)
	$\displaystyle\mathcal{M}^{\mathcal{Y}}_{\mathrm{STO}}(L)$	$\displaystyle=\partial\varphi(\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L)).$		(26)

Since the system evolves only on $\mathcal{M}^{\mathcal{Y}}_{\mathrm{STO}}(L)\in\mathcal{Y}$, the equilibrium state to which the system converges is represented by the intersection:

$\displaystyle\mathcal{M}^{\mathcal{Y}}_{\mathrm{STO}}(L)\cap\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu}),$

(27)

if it is not empty. If the intersection is empty, the system never converges to the equilibrium state and the growth of the volume may happen.

In this subsection, we state our two main claims of this work before presenting their derivations in the following sections.

Our first claim provides the conditions to determine the fate of the system, i.e., growth, shrinking or equilibration. To formulate our claim, we define $y^{\mathrm{min}}$ as

$\displaystyle y^{\mathrm{min}}:=\arg\min_{y}\Set{\varphi^{*}(y)}{y\in\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})}.$

(28)

Note that the productivity of $S$ in Eq. (7) guarantees the existence and the uniqueness of $y^{\mathrm{min}}$ (see the latter half of Appendix F). This $y^{\mathrm{min}}$ gives the chemical equilibrium state whose pressure $\varphi^{*}(y^{\mathrm{min}})$ is the minimum in the candidates, $\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})$.

Here, $L=0$ represents the case in which $L^{l}=0$ for all the components of $L=\{L^{l}\}$ in Eq. (6), and $L\neq 0$ indicates that at least one of the components is not zero.

We first notice that one of the conditions in the claim is given by the sign of $\varphi^{*}(y^{\mathrm{min}})-\tilde{\Pi}$. Here, $\varphi^{*}\left(y^{\mathrm{min}}\right)$ represents the minimum pressure of the system within $\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})$ (see Eq. (28)), whereas $\tilde{\Pi}$ is the pressure of the environment. The growth of the system occurs independently of the value of $L$ when $\varphi^{*}(y^{\mathrm{min}})>\tilde{\Pi}$, because the environmental pressure can not prevent the system from growing by counteracting any internal pressures $\varphi^{*}(y)$ for $y\in\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})$.

The equilibration becomes possible only when $\varphi^{*}(y^{\mathrm{min}})-\tilde{\Pi}\leq 0$ is fulfilled. For the singular case of $L=0$, the equilibration occurs only when the minimum internal pressure perfectly balances with the external one, namely, $\varphi^{*}(y^{\mathrm{min}})-\tilde{\Pi}=0$. When $\varphi^{*}(y^{\mathrm{min}})-\tilde{\Pi}<0$, the system shrinks and $X(t)$ approaches $0$ as $t\to\infty$ because the external pressure dominates the internal one at the equilibrium state. For the generic $L\neq 0$, the fate of the system is qualitatively altered. The system equilibrates even if $\varphi^{*}(y^{\mathrm{min}})-\tilde{\Pi}<0$ owing to the non-singular conservation laws, which prevent $X(t)$ from approaching $0$, i.e., shrinking. In fact, the equilibrium state $y\in\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})$ can have a higher pressure in this case than $\varphi^{*}(y^{\mathrm{min}})$, and it balances with the external pressure $\tilde{\Pi}$. In addition, the system grows for $\varphi^{*}(y^{\mathrm{min}})-\tilde{\Pi}=0$.

From claim 1, the system converges to an equilibrium state when (1) $\varphi^{*}(y^{\mathrm{min}})-\tilde{\Pi}=0$ and $L=0$, and (2) $\varphi^{*}(y^{\mathrm{min}})-\tilde{\Pi}<0$ and $L\neq 0$. For these cases, our second claim describes how the equilibrium states appear in the number space $\mathfrak{X}$.

The fates of the system for the singular case of $L=0$ are basically the same as the case that $S$ is regular; no conservation laws exist so that $\dim\mathrm{Ker}[S^{T}]=0$, and $\dim\mathrm{Ker}[S]=0$ from Eq. (5) ³³3Note that regular $S$ is productive in Eq. (7). (see Sec. V.3 for the reason why regular $S$ can be interpreted as $L=0$). Thus, the two claims clarify that the existence of conservation laws is a fundamental determinant of the fate of the growing system. We investigated the case of regular $S$ in our previous work [19] (see also ⁴⁴4In Sec. II of the Supplementary material, we show an example when $S$ is regular in which the dimension of $\mathrm{Ker}[S^{T}]$ is zero and that of $\mathrm{Im}[S]$ is three.). For comparison, we summarize the results here (see the cases for $L=0$ and regular $S$ in Table 1).

For a regular $S$, the candidates of chemical equilibrium states, $\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})$ in Eq. (23), consist of precisely one point $y_{i}^{\mathrm{EQ}}=-\tilde{\mu}_{m}O^{m}_{r}(S^{-1})^{r}_{i}$. Here, we note that $y^{\mathrm{min}}$ is identical with $y^{\mathrm{EQ}}$ by Eq. (28). Then, we obtain

This statement corresponds to claim 1 in [19].

In this subsection, we demonstrate our claims by using the specific reactions in Example 1 with the ideal gas case (see Example 2).

We first calculate the equilibrium manifold $\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})$ in Eq. (23) to obtain $y^{\mathrm{min}}$ in Eq. (28). For the specific reactions in Eq. (9) and the stoichiometric matrices in Eq. (10), the simultaneous equations in Eq. (22) are written as

$\displaystyle\begin{pmatrix}-y_{1}+2y_{2}-y_{3}-\tilde{\mu}_{1}\\ y_{1}-y_{2}+y_{3}+\tilde{\mu}_{2}\end{pmatrix}=\begin{pmatrix}0\\ 0\end{pmatrix}.$

(29)

The solutions $y=\{y_{i}\}$ are expressed as

$\displaystyle\begin{pmatrix}y_{1}\\ y_{2}\\ y_{3}\end{pmatrix}=\begin{pmatrix}-h\\ \tilde{\mu}_{1}-\tilde{\mu}_{2}\\ \tilde{\mu}_{1}-2\tilde{\mu}_{2}+h\end{pmatrix},$

(30)

where $h\in\mathbb{R}$ is the coordinate of $\mathrm{Ker}[S^{T}]$. The set of solutions in Eq. (30) represents the equilibrium manifold $\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})$.

Second, using Eqs (15) and (20), the full grand potential density $\varphi^{*}(y)$ is obtained for the ideal gas or solution as

$\displaystyle\varphi^{*}(y)=R\tilde{T}\sum_{i}e^{\left\{y_{i}-\nu_{i}^{o}\left(\tilde{T}\right)\right\}/R\tilde{T}}+R\tilde{T}\sum_{m}e^{\left\{\tilde{\mu}_{m}-\mu_{m}^{o}\left(\tilde{T}\right)\right\}/R\tilde{T}}.$

(31)

Using the solutions in Eq. (30), $\varphi^{*}(y)$ is represented on $\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})$ as

	$\displaystyle\bar{\varphi}^{*}(h)$	$\displaystyle=R\tilde{T}\biggl{[}e^{\left\{-h-\nu_{1}^{o}\left(\tilde{T}\right)\right\}/R\tilde{T}}+e^{\left\{\tilde{\mu}_{1}-\tilde{\mu}_{2}-\nu_{2}^{o}\left(\tilde{T}\right)\right\}/R\tilde{T}}$
		$\displaystyle+e^{\left\{\tilde{\mu}_{1}-2\tilde{\mu}_{2}+h-\nu_{3}^{o}\left(\tilde{T}\right)\right\}/R\tilde{T}}\biggr{]}$
		$\displaystyle+R\tilde{T}\left[e^{\left\{\tilde{\mu}_{1}-\mu_{1}^{o}\left(\tilde{T}\right)\right\}/R\tilde{T}}+e^{\left\{\tilde{\mu}_{2}-\mu_{2}^{o}\left(\tilde{T}\right)\right\}/R\tilde{T}}\right].$		(32)

In Fig. 2(a), we numerically plot the pressure $\bar{\varphi}^{*}(h)$ on $\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})$. Here, we have a unique $h^{\mathrm{min}}=(-\nu^{o}_{1}+\nu^{o}_{3}-\tilde{\mu}_{1}+2\tilde{\mu}_{2})/2$ that attains the minimum $\bar{\varphi}^{*}(h)$. Substituting it into Eq. (30), we obtain $y^{\mathrm{min}}$ and the minimum pressure $\varphi^{*}(y^{\mathrm{min}})$. With the parameters given in the caption, $\varphi^{*}(y^{\mathrm{min}})=14.25$.

To demonstrate that the fate of the system is classified by $y^{\mathrm{min}}$ and the conserved quantities $L$ in claim 1, we show numerical simulations of the dynamics, Eq. (3). From the solution $X(t)$, the volume $\Omega(X)$ of the system is determined by the equation of state, Eq. (17). To numerically solve Eq. (3) with the stoichiometric matrix $S$ in Eq. (10), we employ the mass-action kinetics with the local detailed balance condition to specify the flux function $J(t)$ (see Appendix B).

Claim 1: In Fig. 2(b-d), we show numerical trajectories of the volume $\Omega(X)$ from three initial conditions for different pressures $\tilde{\Pi}$. The color of curves corresponds to each initial condition. The fate of the system changes with $\tilde{\Pi}$ as follows. When $\tilde{\Pi}=13<\varphi^{*}(y^{\mathrm{min}})=14.25$ (Fig. 2(b)), the volume of the system increases with time from any initial conditions. As $\tilde{\Pi}$ increases to $\tilde{\Pi}=\varphi^{*}(y^{\mathrm{min}})=14.25$ (Fig. 2(c)), the trajectory from an initial condition with $L=0$ in light blue achieves an equilibrium state, whereas trajectories from the other two with $L\neq 0$ continue growing. As $\tilde{\Pi}$ increases further, $\tilde{\Pi}=16>\varphi^{*}(y^{\mathrm{min}})=14.25$ (Fig. 2(d)), the trajectory with $L=0$ in light blue decreases and the system shrinks. By contrast, those from the other two with $L\neq 0$ achieve equilibrium states. These numerical results demonstrate the claim 1.

Claim 2: In Fig. 2(e,f), we demonstrate the time evolution of the number $X$. When $\tilde{\Pi}=\varphi^{*}(y^{\mathrm{min}})=14.25$ and $L=0$, the equilibrium state to which the system converges depends on the initial conditions in $\mathcal{M}^{\mathfrak{X}}_{\mathrm{STO}}(L=0)$. Furthermore, it indeed lies on the ray $\mathfrak{r}^{\mathcal{Y}}(y^{\mathrm{min}})$, which is given by Eq. (21) (see Fig. 2(e)). In our numerical simulation, we compute it as follows. From $y^{\mathrm{min}}$, we obtain the corresponding density $x^{i}_{\mathrm{min}}=\partial^{i}\varphi^{*}(y^{\mathrm{min}})=\exp\{(y^{\mathrm{min}}_{i}-\nu_{i}^{o})/R\tilde{T}\}$ by using Eq. (31). The ray $\mathfrak{r}^{\mathcal{Y}}(y^{\mathrm{min}})$ is plotted by the one which includes $x_{\mathrm{min}}$. When $\tilde{\Pi}=16>\varphi^{*}(y^{\mathrm{min}})$ and $L=52\neq 0$, the system indeed converges to the unique equilibrium state, irrespective of the initial conditions in $\mathcal{M}^{\mathfrak{X}}_{\mathrm{STO}}(L=52)$ (see Fig. 2(f)).

The fates of the system stated in claims 1 and 2, and Corollary 1 are summarized in Table 1. Before closing this section, we outline their derivations, which will be presented in the subsequent sections. They will be derived in two steps. In the first step, we investigate the existence of an equilibrium state, that is the intersection $\mathcal{M}^{\mathcal{Y}}_{\mathrm{STO}}(L)\cap\mathcal{M}^{\mathcal{Y}}_{\mathrm{EQ}}(\tilde{\mu})$ in Eq. (27). In Sec. III, we will show that the intersection is not empty in the case $2$ for $L=0$ and in the case $3$ for $L\neq 0$ of claim 1, and is empty for the other cases. This will be summarized in Theorem 1. Following Theorem 1, we derive claim 2 from the correspondence in Eq. (21) between the number space $\mathfrak{X}$ and the chemical potential space $\mathcal{Y}$. In the second step, when the intersection is empty, we investigate the landscape of the total entropy function $\Sigma^{\mathrm{tot}}$ to classify whether the system grows or shrinks in Sec. IV. We will show that the system has two possibilities for $L=0$: it grows when $\Sigma^{\mathrm{tot}}$ is not bounded above, or it shrinks when the supremum of $\Sigma^{\mathrm{tot}}$ is at the origin $X=0$. By contrast, the system always grows for $L\neq 0$ when the intersection is empty because $\Sigma^{\mathrm{tot}}$ is not bounded above. This will be summarized in Theorem 2. Mathematical proofs of Theorems 1 and 2, and Corollary 1 will be presented in Sec. V.

The thermodynamic property of the system is encoded in the full grand potential density $\varphi^{*}(y)$ (see Eq. (20)). Here, $y\in\mathcal{Y}=\mathbb{R}^{\mathcal{N}_{X}}$ is the chemical potential of the confined chemicals, and $\varphi^{*}(y)$ represents the pressure of the system at the state $y$. In this work, we assume that $\varphi^{*}(y)$ is a smooth and strictly convex function on $\mathcal{Y}=\mathbb{R}^{\mathcal{N}_{X}}$, and the image of the associated Legendre transformation,

$\displaystyle\partial\varphi^{*}(y):y\in\mathcal{Y}\mapsto\partial\varphi^{*}(y)=\left\{\partial^{i}\varphi^{*}\right\}=\left\{\frac{\partial\varphi^{*}}{\partial y_{i}}\right\},$

(33)

is equal to $\mathcal{X}=\mathbb{R}_{>0}^{\mathcal{N}_{X}}$. In addition, we assume the following properties; (1) $\varphi^{*}(y)$ strictly increases with $y_{i}$ for an arbitrary fixed $\{y_{j}\}_{j\neq i}$. (2) $\partial^{i}\varphi^{*}(y)\rightarrow 0$ when $y_{i}\rightarrow-\infty$. They are satisfied in most thermodynamic systems such as the ideal gas in Eq. (31).

Furthermore, we assume that the mapping $\partial\varphi^{*}:\mathbb{R}^{\mathcal{N}_{X}}\rightarrow\mathbb{R}_{>0}^{\mathcal{N}_{X}}$ is bijective, and we consider that its inverse map is given by $\partial\varphi:\mathbb{R}_{>0}^{\mathcal{N}_{X}}\rightarrow\mathbb{R}^{\mathcal{N}_{X}}$ (see Eq. (13) for $\varphi(x)$). By using this, the above property (2) is rephrased as $y_{i}=\partial_{i}\varphi(x)\rightarrow-\infty$ when $x^{i}\rightarrow 0$.

Finally, we assume that the pressure of the environment $\tilde{\Pi}$ is greater than $\Pi_{\mathrm{min}}$, to guarantee that the volume $\Omega(X)$ is uniquely determined (see Appendix C). Here, $\Pi_{\mathrm{min}}$ denotes the minimum pressure that the system can take as

$\displaystyle\Pi_{\mathrm{min}}:=\inf_{y\in\mathcal{Y}}\varphi^{*}(y)=\lim_{\{y_{i}\}\rightarrow\{-\infty\}}\varphi^{*}(y).$

(34)

The second equality holds from the above property (1).

We remind that the density $x$ is given by the mapping $\rho_{\mathcal{X}}$ in Eq. (18):

$\displaystyle\rho_{\mathcal{X}}:=X\in\mathcal{X}\mapsto\rho_{\mathcal{X}}(X)=x(X)=\left\{\frac{X^{i}}{\Omega(X)}\right\}\in\mathcal{X}.$

(35)

The range of the mapping $\rho_{\mathcal{X}}$ describes possible states of the density $x$ under the isobaric condition. To calculate the range, we compute the volume $\Omega$ by solving the minimization in Eq. (14). Its critical equation is given by

$\displaystyle\varphi\left(\frac{X}{\Omega}\right)-\frac{X^{i}}{\Omega}\partial_{i}\varphi\left(\frac{X}{\Omega}\right)+\tilde{\Pi}=0.$

(36)

Therefore, the possible region in $\mathcal{X}$ under the isobaric condition is constrained in

$\displaystyle\mathcal{I}^{\mathcal{X}}\left(\tilde{\Pi},\tilde{\mu}\right):=\Set{x}{\varphi\left(x\right)-x^{i}\partial_{i}\varphi(x)+\tilde{\Pi}=0,x>0},$

(37)

which we call the isobaric manifold. Since the range of $\rho_{\mathcal{X}}$ is represented in Eq. (37), we find that the mapping $\rho_{\mathcal{X}}(X)$ is a projection of $X\in\mathfrak{X}$ into $\mathcal{I}^{\mathcal{X}}(\tilde{\Pi},\tilde{\mu})$ along the ray which includes $X$.

Example 3: For the ideal gas case, by substituting $\varphi(x)$ in Eq. (15) into Eq. (37), we obtain

$\displaystyle\mathcal{I}^{\mathcal{X}}\left(\tilde{\Pi},\tilde{\mu}\right)=\Set{x}{\sum_{i}x^{i}=\frac{\tilde{\Pi}}{R\tilde{T}}-\sum_{m}\tilde{n}^{m},x>0},$

(38)

where $\tilde{n}^{m}$ is the density of the open chemicals in the environment (see Eq. (16)). Thus, the isobaric manifold in $\mathcal{X}$ represents a simplex for the ideal gas case (Fig. 3(a)).

The mapping $\rho_{\mathcal{X}}$ is the projection of $X\in\mathfrak{X}=\mathbb{R}^{\mathcal{N}_{\mathcal{X}}}_{>0}$ into this simplex (see Fig. 3(b)).

$\blacksquare$

Finally, by using the mapping $\partial\varphi$, the isobaric manifold in $\mathcal{Y}$ is represented as

$\displaystyle\mathcal{I}^{\mathcal{Y}}\left(\tilde{\Pi},\tilde{\mu}\right):=\partial\varphi\left(\mathcal{I}^{\mathcal{X}}\right)=\Set{y}{\varphi^{*}\left(y\right)-\tilde{\Pi}=0},$

(39)

which is straightforward to interpret that the pressure of the system $\varphi^{*}(y)$ at the chemical potential $y$ is balanced with $\tilde{\Pi}$ in this manifold.

The accessible subspace of the state $X(t)$ from an initial condition $X_{0}$ in Eq. (4) is

$$\mathcal{M}^{\mathfrak{X}}_{\mathrm{STO}}(L):=\Set{X}{U^{l}_{i}X^{i}=U^{l}_{i}X^{i}_{0}=L^{l},X>0},$$

(40)

which we call the stoichiometric manifold in $\mathfrak{X}$.

By applying the mapping $\rho_{\mathcal{X}}$ to $\mathcal{M}^{\mathfrak{X}}_{\mathrm{STO}}(L)$, we have the stoichiometric manifold in the density space $\mathcal{X}$ as

	$\displaystyle\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L):=$	$\displaystyle\left.\biggl{\{}x\right|\varphi(x)-x^{i}\partial_{i}\varphi(x)+\tilde{\Pi}=0,$
		$\displaystyle U^{l}_{i}x^{i}=\frac{L^{l}}{\alpha},x>0\biggr{\}},$		(41)

where $\alpha>0$ is arbitrary. Using the mapping $\partial\varphi$, the stoichiometric manifold in the chemical potential space $\mathcal{Y}$ is given by

$$\mathcal{M}^{\mathcal{Y}}_{\mathrm{STO}}(L):=\Set{y}{\varphi^{*}(y)-\tilde{\Pi}=0,U^{l}_{i}\partial^{i}\varphi^{*}(y)=\frac{L^{l}}{\alpha}}.$$

(42)

Example 4: For the specific stoichiometric matrix in Eq. (10), $\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L)$ is schematically depicted in Fig. 3(c,d) by applying the mapping $\rho_{\mathcal{X}}$ to $\mathcal{M}^{\mathfrak{X}}_{\mathrm{STO}}(L)$.

$\blacksquare$

For further investigations, we define the following manifold in $\mathcal{X}$ as

$\displaystyle\hat{\mathcal{M}}^{\mathcal{X}}_{\mathrm{STO}}\left(\hat{L}\right):=\Set{x}{U^{l}_{i}x^{i}=\hat{L}^{l},x>0},$

(43)

and that in $\mathcal{Y}$ as

	$\displaystyle\hat{\mathcal{M}}^{\mathcal{Y}}_{\mathrm{STO}}\left(\hat{L}\right)$	$\displaystyle:=\partial\varphi\left(\hat{\mathcal{M}}^{\mathcal{X}}_{\mathrm{STO}}\left(\hat{L}\right)\right)$
		$\displaystyle=\Set{y}{U^{l}_{i}\partial^{i}\varphi^{*}(y)=\hat{L}^{l}}.$		(44)

These manifolds correspond to the stoichiometric manifolds with conserved quantities $\hat{L}$ under the isochoric condition [78]. Hereafter, we term the isochoric stoichiometric manifolds.

Using these manifolds, the stoichiometric manifold under isobaric condition is represented as follows:

$\displaystyle\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L)=\bigcup_{\alpha>0}\hat{\mathcal{M}}^{\mathcal{X}}_{\mathrm{STO}}\left(\frac{L}{\alpha}\right)\cap\mathcal{I}^{\mathcal{X}}(\tilde{\Pi},\tilde{\mu}),$

(45)

and

$\displaystyle\mathcal{M}^{\mathcal{Y}}_{\mathrm{STO}}(L)=\bigcup_{\alpha>0}\hat{\mathcal{M}}^{\mathcal{Y}}_{\mathrm{STO}}\left(\frac{L}{\alpha}\right)\cap\mathcal{I}^{\mathcal{Y}}(\tilde{\Pi},\tilde{\mu}).$

(46)

For arbitrary $c>0$, we find $\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(cL)=\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L)$ and, correspondingly, $\mathcal{M}^{\mathcal{Y}}_{\mathrm{STO}}(cL)=\mathcal{M}^{\mathcal{Y}}_{\mathrm{STO}}(L)$. Hence, each ray $\mathfrak{r}$ in the space $\mathcal{L}$ of conserved quantities characterizes these manifolds (see also Example 5 below and Fig. 3(d-f)). To be more precise, for any $L,L^{\prime}\in\mathfrak{r}$, $\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L)=\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L^{\prime})$ holds. We denote the stoichiometric manifold corresponding to a ray $\mathfrak{r}$ as $\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L\in\mathfrak{r})$.

When $L=0$, the stoichiometric manifold in $\mathcal{X}$ is written as

$\displaystyle\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L=0)=\hat{\mathcal{M}}^{\mathcal{X}}_{\mathrm{STO}}\left(\hat{L}=0\right)\cap\mathcal{I}^{\mathcal{X}}(\tilde{\Pi},\tilde{\mu}).$

(47)

Since the dimension of $\alpha$ in Eq. (45) vanishes for $\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L=0)$, we obtain $\dim\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L=0)=\dim\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L\neq 0)-1$. In addition, for any given $L\neq 0$, we find that $\hat{\mathcal{M}}^{\mathcal{X}}_{\mathrm{STO}}\left(L/\alpha\right)\rightarrow\hat{\mathcal{M}}^{\mathcal{X}}_{\mathrm{STO}}(\hat{L}=0)$ when $\alpha\rightarrow\infty$. Thus, the stoichiometric manifold $\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L=0)$ forms an interface of all the stoichiometric manifolds $\mathcal{M}^{\mathcal{X}}_{\mathrm{STO}}(L\neq 0)$. These properties hold similarly in $\mathcal{Y}$ by replacing the superscript $\mathcal{X}$ with $\mathcal{Y}$.