# Is Independence Necessary for a Discontinuous Phase Transition within the q-Voter Model?

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Results

#### 3.1. Time Evolution

#### 3.2. Stationary States

**The lower spinodal as a function of parameters ${\mathit{q}}_{\mathit{a}},{\mathit{q}}_{\mathit{c}}$**. It corresponds to the value of $p={p}_{1}^{\ast}$ at ${c}_{st}=1/2$, so it can be easily calculated from the relation $p=p({c}_{st})$ given by Equation (13). In the case of a continuous phase transition this is simply the critical point ${p}^{\ast}={p}_{1}^{\ast}$, which separates two phases. In this case, it corresponds to the maximum of $p=p({c}_{st})$, whereas in case of a discontinuous phase transition it corresponds to the minimum of $p=p({c}_{st})$, as seen in Figure 2 and Figure 3. As already written, $p={p}_{1}^{\ast}$ is also a pitchfork bifurcation point, given by Equation (12), at which a steady state $c=1/2$ changes stability, i.e., for $p<{p}_{1}^{\ast}$ an agreement phase ($c\ne 1/2$) is stable and disagreement phase ($c=1/2$) is unstable. This indicates that independently on the initial state of the system an agreement phase is reached. Although ${p}_{1}^{\ast}$ has been already calculated within linear stability analysis, we will show that indeed it can be also obtained from Equation (13).**The tricritical point, i.e., the value of ${\mathit{q}}_{\mathit{c}}={\mathit{q}}_{\mathit{c}}^{\mathbf{\ast}}$ as a function of ${\mathit{q}}_{\mathit{a}}$ for which the transition switches from continuous to discontinuous.**As described above at this point the minimum at ${c}_{st}=1/2$ changes to maximum and thus this point can be also easily derived by calculating the point in which the second derivative of p changes the sign.**The upper spinodal $\mathit{p}={\mathit{p}}_{\mathbf{2}}^{\ast}$ as a function of parameters ${\mathit{q}}_{\mathit{a}},{\mathit{q}}_{\mathit{c}}$.**As written above, in case of discontinuous phase transition $p=p({c}_{st})$ has two maxima at ${c}_{+}$ and ${c}_{-}$ and the value $p({c}_{+})=p({c}_{-})={p}_{2}^{\ast}$ is the upper spinodal, so it can be also derived from the relation $p=p({c}_{st})$ given by Equation (13). In theory calculations are straightforward. Unfortunately, it occurs that finding an analytical formula for ${p}_{2}^{\ast}={p}_{2}^{\ast}({q}_{a},{q}_{c})$ for arbitrary values of parameters ${q}_{a}$ and ${q}_{c}$ is impossible and the upper spinodal will be obtained numerically.**The point of the phase transition ${\mathit{p}}^{\ast}={\mathit{p}}^{\ast}({\mathit{q}}_{\mathit{a}},{\mathit{q}}_{\mathit{c}})$.**For a continuous phase transition, it is straightforward, as described above, because it corresponds to the value of $p={p}_{1}^{\ast}$ at ${c}_{st}=1/2$. In fact, for a continuous phase transition all three points: lower spinodal ${p}_{1}^{\ast}$, upper spinodal ${p}_{2}^{\ast}$ and the point of the phase transition ${p}^{\ast}$ collapse to the single critical point, i.e., ${p}^{\ast}={p}_{1}^{\ast}={p}_{2}^{\ast}$. For a discontinuous phase transition, it is far less trivial. The transition point is placed between lower and upper spinodals. In thermodynamics it corresponds to the point, at which phases are in the equilibrium, i.e., corresponding thermodynamic potential has minima of equal depth. Here we will also introduce an equivalent of a potential and use it to calculate the transition point.

#### 3.3. Landau Approach for Continuous and Discontinuous Phase Transitions

**the Landau theory describes continuous phase transitions only for $\mathit{B}>\mathbf{0}$**. In Figure 4 we present a potential, given by Equation (20), for $B>0$ (specifically for $B=1$) and three values of A: $A=-1$, $A=0$ and $A=1$. It is seen that indeed for $A<0$ the potential has three extrema: maximum at $m=0$ and two minima corresponding to ${m}^{2}=\pm -A/2B$. For $A>0$ the potential has only one extremum: minimum at $m=0$. This means that for $A=0$ the steady state $m=0$ loses stability. In the next section we will show that indeed this condition is equivalent with the condition given by Equation (12).

**for $\mathit{B}<\mathbf{0},\mathit{C}>\mathbf{0}$ we can describe discontinuous phase transitions**[40]. At the end of this section we will see why the assumption $C>0$ is needed. For $B<0$ we have to take into account the next term of the power series:

#### 3.4. Application of the Landau Approach

- The critical point $p={p}_{1}^{\ast}$ at which solution $m=0(c=1/2)$ loses stability corresponds to $A=0$.
- For $B=0$ there is a tricritical point at $A=0$, which means that for $B>0$ the transition is continuous, whereas for $B<0$ it is discontinuous.
- For $B>0$ the transition is continuous, see Figure 6. In such a case potential takes one of two forms. For $p<{p}_{1}^{\ast}$ potential V is a double-well one with maximum at $m=0$ ($c=1/2$). It means that the system always reaches one of two ordered phases: it is attracted by the minima of V and repelled by the maximum at $m=0$, which corresponds to the unstable fixed point. For $p>{p}_{1}^{\ast}$ the potential has only one minimum that corresponds to $m=0$ ($c=1/2$). It means that a system always reaches disordered phase, i.e., the fixed point $m=0$ ($c=1/2$) is stable.
- For $B<0$ the phase transition is discontinuous, see Figure 7, and we can calculate the transition point ${p}^{\ast}$ from the condition (35) as well as spinodals ${p}_{1}^{\ast},{p}_{2}^{\ast}$ from (31). As previously, the potential has two minima (ordered state) and one maximum (disordered state) for $p<{p}_{1}^{\ast}$, i.e., below lower spinodal. For $p\in ({p}_{1}^{\ast},{p}_{2}^{\ast})$ the potential has five extrema: two maxima corresponding to unstable fixed points and three minima corresponding to stable fixed points. Finally, for $p>{p}_{2}^{\ast}$ the potential has only one minimum that corresponds to $m=0$ ($c=1/2$).

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Average trajectories for ${q}_{a}=2$, two values of ${q}_{c}$ (${q}_{c}=4$ in the upper and ${q}_{c}=8$ in the bottom panels) and several values of p increasing from left to right. Analytical results are marked by the solid lines, whereas Monte Carlo results are marked by symbols. Simulation results were obtained for the graph of size $N={10}^{4}$ and averaged over 100 samples. The area between the bounded lines showing the standard deviation is marked by the light blue color. Exact values of p from left to right are the following: $p=0.18,0.2,0.22$ (upper panels, i.e., in for ${q}_{a}=2,{q}_{c}=4$) and $p=0.03,0.05,0.07$ (bottom panels, i.e., for ${q}_{a}=2,{q}_{c}=8$).

**Figure 2.**Flow diagrams for ${q}_{a}=2$ and two values of ${q}_{c}$: ${q}_{c}=4$ (left panel) and ${q}_{c}=8$ (right panel), which are the same values as in Figure 1. Here solid lines denote stable steady values of concentration ${c}_{st}$, whereas dashed lines denote unstable values of ${c}_{st}$. Arrows denote the direction of flow, i.e., how the concentration changes in time. Supercritical pitchfork bifurcation that corresponds to the continuous phase transition is seen in the left panel, whereas subcritical pitchfork bifurcation that corresponds to the discontinuous phase transition is seen in the right panel [28,39].

**Figure 3.**Dependence between the stationary concentration of positive opinions ${c}_{st}$ and the probability of anticonformity p for ${q}_{a}=2$ (

**left panel**) and ${q}_{a}=4$ (

**right panel**).

**Figure 4.**Potential given by Equation (20) for $B>0(B=1)$ and three values of A: $A<0(A=-1)$ (

**left panel**), $A=0$ (

**middle panel**) and $A>0(A=1)$ (

**right panel**).

**Figure 5.**Potential given by Equation (24) for $B<0(B=-1)$, $C>0(C=1)$ and three values of A: $A=0$, which corresponds to lower spinodal (

**left panel**), $A={B}^{2}/4C$, which corresponds to the point of the phase transition (

**middle panel**) and $A={B}^{2}/3C$, which corresponds to upper spinodal (

**right panel**).

**Figure 6.**Potential for ${q}_{a}=2,{q}_{c}=4$ and several values of the probability of anticonformity. From left to right: $p<<{p}^{\ast},p<{p}^{\ast},p={p}^{\ast},p>{p}^{\ast},p>>{p}^{\ast}$, where ${p}^{\ast}=0.2$ and exact values of p from left to right are the following: $p=0.05,0.15,0.2,0.25,0.35$.

**Figure 7.**Potential for ${q}_{a}=2,{q}_{c}=8$ and several values of the probability of anticonformity. From left to right: $p<{p}_{1}^{\ast},p\in ({p}_{1}^{\ast},{p}^{\ast}),p={p}^{\ast},p\in ({p}^{\ast},{p}_{2}^{\ast}),p>{p}^{\ast}$, where ${p}^{\ast}=0.0564$ and exact values of p from left to right are the following: $p=0.02,0.05,0.0564,0.06,0.09$.

**Figure 8.**Phase diagrams for ${q}_{a}=2$ (

**left panel**) and ${q}_{a}=4$ (

**right panel**). Points of continuous phase transitions are marked by ∘ and discontinuous by ∗. Solid lines without symbols denote spinodal lines, i.e., limits of the region with metastability, in which the final state depends on the initial one.

**Figure 9.**

**Left panel**: the critical point ${p}_{1}^{\ast}$, below which the system is ordered ($m\ne 0$) independently on the initial state, as a function of ${q}_{a}$ and ${q}_{c}$.

**Right panel**: the width of hysteresis ${p}_{2}^{\ast}-{p}_{1}^{\ast}$ as a function of ${q}_{a}$ and ${q}_{c}$.

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**MDPI and ACS Style**

Abramiuk, A.; Pawłowski, J.; Sznajd-Weron, K.
Is Independence Necessary for a Discontinuous Phase Transition within the *q*-Voter Model? *Entropy* **2019**, *21*, 521.
https://doi.org/10.3390/e21050521

**AMA Style**

Abramiuk A, Pawłowski J, Sznajd-Weron K.
Is Independence Necessary for a Discontinuous Phase Transition within the *q*-Voter Model? *Entropy*. 2019; 21(5):521.
https://doi.org/10.3390/e21050521

**Chicago/Turabian Style**

Abramiuk, Angelika, Jakub Pawłowski, and Katarzyna Sznajd-Weron.
2019. "Is Independence Necessary for a Discontinuous Phase Transition within the *q*-Voter Model?" *Entropy* 21, no. 5: 521.
https://doi.org/10.3390/e21050521