Entropy of structure-forming systems

To calculate the entropy of structure-forming systems, we first define a set of possible microstates and mesostates. Let us consider a system of n particles. Each single particle can attain states from the set $$ {\mathcal{X}}^{\left(1\right)}=\left\{x_1^{\left(1\right)},\dots ,x_{m_1}^{\left(1\right)}\right\}$$. The superscript number (1) indicates that the states correspond to a single-particle state, and m₁ denotes the number of these states. A typical set of states could be the spin of the particle {↑,↓}, or a set of energy levels. Having only single-particle states, the microstate of the system consisting of n particles is a vector (X₁, X₂, …, X_n), where $$ X_k\in {\mathcal{X}}^{\left(1\right)}$$ is the state of kth particle. Let us now assume that any two particles can create a two-particle state. This two-particle state can be a molecule composed of two atoms, a cluster of two colloidal particles, etc. We call this state as a cluster. This two-particle cluster can attain states $$ {\mathcal{X}}^{\left(2\right)}=\left\{x_1^{\left(2\right)},\dots ,x_{m_2}^{\left(2\right)}\right\}$$. A microstate of a system of n particles is again a vector (X₁, X₂, …, X_n), but now either $$ X_k\in {\mathcal{X}}^{\left(1\right)}$$ or $$ X_k\in {\mathcal{X}}^{\left(2\right)}\times {\mathbb{Z}}_n^2$$. For instance, a state of particle k belonging to a two-particle cluster can be written as $$ X_k=x_1^{\left(2\right)}\left(k_1,k_2\right)$$. The indices in the brackets tell us that the particle k belongs to the cluster of size two in the state $$ x_1^{\left(2\right)}$$ and the cluster is formed by particles k₁ and k₂ (k₁ < k₂). Indeed, either k₁ = k or k₂ = k.

Now assume that particles can also form larger clusters up to a maximal size, m. Consider m as a fixed number, m ≤ n. Generally, clusters of size j have states $$ {\mathcal{X}}^{\left(j\right)}=\left\{x_1^{\left(j\right)},\dots ,x_{m_j}^{\left(j\right)}\right\}$$. The corresponding states of the particle are always elements from sets $$ {\mathcal{X}}^{\left(j\right)}\times {\mathbb{Z}}_n^j$$ with the restriction that if the kth particle is in a state $$ x_i^{\left(j\right)}\left(k_1,\dots ,k_j\right)$$ then k_l < k_l+1, for all l and one k_l = k. Consider an example of four particles. Particles are either in a free state or they form a cluster of size two. A state of each particle is either s⁽¹⁾—a free particle, or x⁽²⁾(i, j)—a cluster compound from particles i and j. As an example, a typical microstate is Ψ = (x⁽¹⁾, x⁽²⁾(2,3), x⁽²⁾(2,3), x⁽¹⁾), which means that particles 1 and 4 are free and particles 2 and 3 form a cluster.

Now consider a mesoscopic scale, where the mesostate of the system is given only by the number of clusters in each state $$ x_i^{\left(j\right)}$$. Let us denote $$ n_i^{\left(j\right)}$$ as the number of clusters in state $$ x_i^{\left(j\right)}$$. The mesostate is therefore characterized by a vector $$ \mathbb{N}=\left(n_i^{\left(j\right)}\right)$$, which corresponds to a frequency (histogram) of microstates. The normalization condition is given by the fact that the total number of particles is n, i.e., $$ {\sum }_{ij}j{n}_i^{\left(j\right)}=n$$. For example, a mesostate, $$ {\mathbb{N}}_{\mathrm{\Psi }}$$, corresponding to a microstate Ψ is $$ {\mathbb{N}}_{\psi }=\left(n^{\left(1\right)}=2,n^{\left(2\right)}=1\right)$$, denoting that there are two free particles and one two-particle cluster.

The Boltzmann entropy³⁶ of this mesostate is given by

The number of microstates giving the same mesostate can be expressed as the product of configurations with the same state for each $$ x_i^{\left(j\right)}$$. Let us begin with the particles that do not form clusters. The number of equivalent representations for one distinct state is $$ \left(n_i^{\left(1\right)}\right)!$$, which corresponds to the number of permutations of all particles in the same state. For the cluster states, one can think about equivalent representations of one microstate in two steps: first permute all clusters, which gives $$ \left(n_i^{\left(j\right)}\right)!$$ possibilities. Then, permute the particles in the cluster, which gives j! possibilities for every cluster, so that we end up with $$ {\left(j!\right)}^{n_i^{\left(j\right)}}$$ combinations.

As an example, consider the case of four particles. First, we look at free particles that attain states $$ x_1^{\left(1\right)}$$ or $$ x_2^{\left(1\right)}$$. Let us consider a mesostate $$ {\mathbb{N}}_1=\left(n_1^{\left(1\right)}=2,n_2^{\left(1\right)}=2\right)$$, i.e., two particles in the first state and two particles in the second. The number of distinct microstates corresponding to the mesostate $$ {\mathbb{N}}_1$$ is given by $$ W\left({\mathbb{N}}_1\right)=4!/\left(2!2!\right)=6$$. All microstates that belong to the mesostate $$ {\mathbb{N}}_1$$ are

For example, a microstate (x⁽²⁾(2, 1), x⁽²⁾(2, 1), x⁽²⁾(4, 3), x⁽²⁾(4,3)) is the same as the first microstate because we just relabel 1↔2 and 3↔4. In summary, the multiplicity corresponding to $$ x_i^{\left(j\right)}$$ is $$ \left(n_i^{\left(j\right)}\right)!{\left(j!\right)}^{n_i^{\left(j\right)}}$$, and we can express the total multiplicity as

In the remainder, we denote thermodynamic quantities per particle by calligraphic script and total quantities by normal script. We express the entropy per particle as

Finite interaction range

Up to now, we assumed an infinite range of interaction between particles, which is unrealistic for chemical reactions, where only atoms within a short range form clusters. A simple correction is obtained by dividing the system into a fixed number of boxes: particles within the same box can form clusters, particles in different boxes cannot. We begin by calculating the multiplicity for two boxes. For simplicity, assume that they both contain n/2 particles. The multiplicity of a system with two boxes, $$ \tilde{W}\left(n_i^{\left(j\right)}\right)$$, is given by the sum of all possible partitions of $$ n_i^{\left(j\right)}$$ clusters with state $$ x_i^{\left(j\right)}$$ into the first box (containing $$ {}^1{n}_i^{\left(j\right)}$$ clusters) and the second box (containing $$ {}^2{n}_i^{\left(j\right)}$$ clusters), such that $$ n_i^{\left(j\right)}={}^1{n}_i^{\left(j\right)}+{}^2{n}_i^{\left(j\right)}$$. The multiplicity is therefore

where W is the multiplicity in Eq. (2). The dominant contribution to the sum comes from the term, where $$ {}^1{n}_i^{\left(j\right)}={}^2{n}_i^{\left(j\right)}=n_i^{\left(j\right)}/2$$, so that we can approximate the multiplicity by $$ \tilde{W}\left(n_i^{\left(j\right)}\right)\approx W{\left(n_i^{\left(j\right)}/2\right)}^2$$. Similarly, for b boxes we obtain the multiplicity

By defining the concentration of particles as $$ c=n/b$$, the entropy per particle becomes

or, respectively,

Note that the entropy of structure-forming systems is both additive and extensive in the sense of Lieb and Yngvason³⁷. It is also concave, ensuring the uniqueness of the maximum entropy principle. For more details and connections to axiomatic frameworks, see Supplementary Discussion.

Equilibrium thermodynamics of structure-forming systems

We now focus on the equilibrium thermodynamics obtained, for example, by considering the maximum entropy principle. Consider the internal energy

Comparison with the grand-canonical ensemble

To compare the presented exact approach with the grand-canonical ensemble, consider the simple chemical reaction, 2X⇌X₂. Without loss of generality, assume that free particles carry some energy, ϵ. We calculate the Helmholtz free energy for both approaches in Supplementary Information. In Fig. 1, we show the corresponding specific heat, $$ c\left(T\right)=-T\frac{{\partial }^2\mathcal{F}}{\partial T^2}$$. For large systems, the usual grand-canonical ensemble approach and the exact calculation with a strictly conserved number of particles converge. For small systems, however, there appear notable differences. This is visible in Fig. 1, where only for large n and low concentrations, $$ c$$, the specific heat for the exact approach (squares) and the grand-canonical ensemble (triangles) become identical. The inset shows the ratio of the specific heat, c_C/c_GC − 1, vanishing for large n. For large systems, the exact approach and the the grand-canonical ensemble are equivalent.

Fig. 1

Specific heat, c(T), for the reaction 2X⇌X₂ for the presented canonical approach with an exact number of particles in comparison to the grand-canonical ensemble.

The specific heat for the canonical ensemble (C) is drawn by squares, and the specific heat for the grand-canonical ensemble (GC) is drawn by triangles. n denotes the number of particles. For small systems the difference of the approaches becomes apparent. The inset shows the ratio of the specific heat calculated from the exact approach to the one obtained from the grand-canonical ensemble, c_C/c_GC − 1. For large n the quantity decays to zero for any temperature.

Relation to the theory of self-assembly

In many applications, the number of energetic configurations for each cluster size is so large that one is only interested in the distribution of cluster sizes. For this case, it is possible to formulate an effective theory considering contributions from all configurations that is known as the theory of self-assembly. For an overview, see Likos et al.²⁷.

To compute the free energy in terms of the cluster-size distribution, we define the latter as

Examples for thermodynamics of structure-forming systems

We now apply the results obtained in the previous section to several examples of structure-forming systems. We particularly focus on how the presence of mescoscopic structures of clustered states leads to the macroscopic physical properties. In the presence of structure formation, there exists a phase transition between a free particle fluid phase and a condensed phase, containing clusters of particles. This phase transition is demonstrated in two examples.

The first example on soft-matter self-assembly describes the process of condensation of one-patch colloidal amphibolic particles. This condensation is relevant in applications in nanomaterials and biophysics. The second example covers the phase transition of the Curie–Weiss spin model for the situation where particles form molecules. In Supplementary Information, we discuss the additional examples of a magnetic gas and a size-dependent chemical potential.

Kern–Frenkel model of patchy particles

Recently, the theory of soft-matter self-assembly has successfully predicted the creation of various structures of colloidal particles, including clusters of Janus particles³², polymerization of colloids³⁸, and the crystallization of multipatch colloidal particles³⁹. Kern and Frenkel⁴⁰ introduced a simple model to describe the self-assembly of amphibolic particles with two-particle interactions. r_ij denotes a unit vector connecting the centers of particles i and j, r_ij is the corresponding distance, and n_i and n_j are unit vectors encoding the directions of patchy spheres. The Kern–Frenkel potential was defined as

where

and

The characteristic quantity, $$ \chi ={\sin }^2\left(\theta /2\right)$$, is the particle coverage. In the theory of self-assembly, the cluster-size distribution is determined by the partial partition functions Eq. (19). Due to the enormous number of possible configurations, it is impossible to calculate $$ {\mathcal{Z}}_j$$ analytically and simulation methods were introduced, including a grand-canonical Monte Carlo method and successive umbrella sampling; for a review, see Rovigatti et al.⁴¹. Instead of calculating the exact value of $$ {\mathcal{Z}}_j$$, we use a stylized model based on Fantoni et al.³². There the partial partition function is parameterized as $$ \frac{\log {\mathcal{Z}}_j}{jϵ}=b\tanh \left(aj\right)$$, where b < 0 and a > 0 are the model parameters. While for small cluster sizes, the free energy per particle decreases linearly with the size, for larger clusters, it saturates at b. To calculate the average cluster size, Eq. (16), one has to solve the equation for Λ, Eq. (15). In Fig. 2, we show the phase diagram of the patchy particles for b = − 3 and a = 25 and n = 100. The average number of clusters, M, plays the role of the order parameter. In the phase diagram, one can clearly distinguish three phases. At high temperature, we observe the liquid phase, where most particles are not bound to others. At low temperatures, we have a condensed phase with macroscopic clusters. The two phases are separated by a coexistence phase, where both large clusters and unbounded particles are present. The coexistence phase (gray region) is characterized by a bimodal distribution that can be recognized by calculating the bimodality coefficient⁴². Results presented in Fig. 2 qualitatively correspond to results obatined in Fantoni et al.³² for the case of Janus particles with χ = 0.5.

Fig. 2

Phase diagram for the self-assembly of patchy particles for n = 100 particles.

The average cluster size (M) as a function of temperature (T) and concentration (c) is seen. The cluster size is given by the color and ranges from M = 0 (purple) to M = 100 (red). We observe three phases: the liquid and condensed phase are divided by a coexistence phase (gray area). Coexistence is characterized by a bimodal distribution that can be detected with a shift in the bimodality coefficient.

Curie–Weiss model with molecule formation

To discuss an example of a spin system with molecule states, consider the fully connected Ising model^43–46 with a Hamiltonian that allows for possible molecule states

Molecule states neither feel the spin–spin interaction nor the external magnetic field, h. Therefore, the sum only extends over free particles. In a mean-field approximation, we use the magnetization, $$ m=\frac{1}{n-1}{\sum }_{i\ne j}{\sigma }_i$$, and express the Hamiltonian as H^MF(σ_i) = −(Jm + h)∑_j,freeσ_j. The self-consistency equation $$ m=-\frac{\partial F}{\partial h}{\mid }_{h=0}$$ leads to an equation for m that is calculated numerically (Supplementary Information) and that is shown in Fig. 3. Contrary to the mean-field approximation of the usual fully connected Ising model (without molecule states), the phase transition is no longer second-order but becomes first-order. There exists a bifurcation where solutions for m = 0 and m > 0 are stable. The second-order transition is recovered for small systems, n→0. The critical temperature is shifted toward zero for increasing n. We performed Monte Carlo simulations to check the result of the mean-field approximation; see Supplementary Information.

Fig. 3

Magnetization of the fully connected Ising model with molecule states for n = 50 and n = 200 particles, for a spin–spin coupling constant, J = 1.

Results of the mean-field approximation (solid lines) are in good agreement with Monte Carlo simulations (symbols). Errorbars show the standard deviation of the average value obtained from 1000 independent runs of the simulations (see Supplementary Information for more details). The inset shows the well-known result for the fully connected Ising model without molecule states. Without molecule formation, we observe the usual second-order transition. With molecules, the critical temperature decreases with the number of particles and the phase transition becomes first-order.

Stochastic thermodynamics of structure-forming systems

Consider an arbitrary nonequilibrium state given by $$ {\wp }_i^{\left(j\right)}\equiv {\wp }_i^{\left(j\right)}\left(t\right)$$, and imagine that the evolution of the probability distribution is defined by a first-order Markovian linear master equation, as is usually assumed in stochastic thermodynamics^47,48

Let us now consider a stochastic trajectory, x(τ) = (i(τ),j(τ)), denoting that at time τ, the particle is in state $$ x_{i\left(\tau \right)}^{\left(j\left(\tau \right)\right)}$$. We introduce the time-dependent protocol, l(τ), that controls the energy spectrum of the system. The stochastic energy for trajectory x(τ) and protocol l(τ) can be expressed as $$ ϵ\left(\tau \right)\equiv {ϵ}_{i\left(\tau \right)}^{\left(j\left(\tau \right)\right)}\left(l\left(\tau \right)\right)$$. We assume microreversibility from which follows that detailed balance is valid even when the energy spectrum is time-dependent (due to protocol l(τ)). We define the stochastic entropy as

The time-reversed trajectory is $$ \tilde{\mathbf{x}}\left(\tau \right)=\left(i\left(T-\tau \right),j\left(T-\tau \right)\right)$$, and the time-reversed protocol is $$ \tilde{l}\left(\tau \right)=l\left(T-\tau \right)$$. The log-ratio of the probability, $$ \mathcal{P}$$, of a forward trajectory and the probability, $$ \tilde{\mathcal{P}}$$, of the time-reversed trajectory under the time-reversed protocol is equal to $$ \mathrm{\Delta }\sigma =\mathrm{\Delta }{s}_i+\log \frac{j_0}{{\tilde{j}}_0}$$, where j₀ = j(τ = 0) and $$ {\tilde{j}}_0=\tilde{j}\left(\tau =0\right)$$, see Supplementary Information. Hence, $$ \log \frac{\mathcal{P}\left(\mathbf{x}\left(\tau \right)\right)}{\tilde{P}\left(\tilde{\mathbf{x}}\left(\tau \right)\right)}=\mathrm{\Delta }\sigma $$, which leads to the fluctuation theorem⁴⁹

If we start in an equilibrium distribution with j(τ = 0) = j₀ and the reverse experiment also starts in an equilibrium distribution with $$ \tilde{j}\left(\tau =0\right)={\tilde{j}}_0$$, by plugging this into Eq. (31) and a simple manipulation, we have