Contact Process with Aperiodic Temporal Disorder

Abstract

We investigate the nonequilibrium critical behavior of the contact process with deterministic aperiodic temporal disorder implemented by choosing healing or infection rates according to a family of aperiodic sequences based on the quasiperiodic Fibonacci sequence. This family allows us to gauge the temporal fluctuations via a wandering exponent $\omega$ and put our work in the context of the Kinzel–Vojta–Dickman criterion for the relevance of temporal disorder to the critical behavior of nonequilibrium models. By means of analytic and numerical calculations, the generalized criterion is tested in the mean-field limit.

Appendices

Appendix A. Properties of the Generalized Fibonacci Sequence

For the generalized Fibonacci sequence defined by the substitution rule $A\rightarrow AB^{k}$ and $B\rightarrow A$, the numbers $N_{A}^{\left( j\right) }$ and $N_{B}^{\left( j\right) }$ of letters A and B in the finite sequence obtained after j iterations of the rule are given by the matrix equation

$$\begin{aligned} \left( \begin{array}{c} N_{A}^{\left(\kern 0.10emj\kern 0.10em\right) }\\ N_{B}^{\left(\kern 0.10emj\kern 0.10em\right) } \end{array}\right) =\varvec{\Omega }^{\kern 0.10emj\kern 0.10em}\left( \begin{array}{c} 1\\ 0 \end{array}\right) , \end{aligned}$$

(37)

in which we assume that the sequence is built starting from a single letter A and $\varvec{\Omega }$ is the substitution matrix

$$\begin{aligned} \varvec{\Omega }=\left( \begin{array}{cc} 1 &{} 1\\ k &{} 0 \end{array}\right) . \end{aligned}$$

(38)

Diagonalizing $\varvec{\Omega }$, we can write

$$\begin{aligned} \varvec{\Omega }=\textbf{U}\left( \begin{array}{cc} \zeta _{+} &{} 0\\ 0 &{} \zeta _{-} \end{array}\right) \textbf{U}^{-1},\qquad \textbf{U}=\left( \begin{array}{cc} \zeta _{-}/k &{} \zeta _{+}/k\\ 1 &{} 1 \end{array}\right) , \end{aligned}$$

(39)

with

$$\begin{aligned} \zeta _{\pm }=\frac{1\pm \sqrt{1+4k}}{2}, \end{aligned}$$

(40)

so that

$$\begin{aligned} \varvec{\Omega }^{\kern 0.10emj}=\textbf{U}\left( \begin{array}{cc} \zeta _{+}^{j} &{} 0\\ 0 &{} \zeta _{-}^{j} \end{array}\right) \textbf{U}^{-1}, \end{aligned}$$

(41)

leading to

$$\begin{aligned} N_{A}^{\left( j\right) }=\frac{\zeta _{+}^{j+1}-\zeta _{-}^{j+1}}{\sqrt{4k+1}},\quad N_{B}^{\left( j\right) }=k\frac{\zeta _{+}^{j}-\zeta _{-}^{j}}{\sqrt{4k+1}}. \end{aligned}$$

(42)

Taking into account that $\zeta _{+}>\left| \zeta _{-}\right|$, the asymptotic fractions of letters A and B are, respectively,

$$\begin{aligned} x_{A}=\lim _{j\rightarrow \infty }\frac{N_{A}^{\left(\kern 0.10emj\kern 0.10em\right) }}{N_{j}}=\frac{1}{\zeta _{+}} \end{aligned}$$

(43)

and

$$\begin{aligned} x_{B}=\lim _{j\rightarrow \infty }\frac{N_{B}^{\left( j\right) }}{N_{j}}=1-\frac{1}{\zeta _{+}}, \end{aligned}$$

(44)

and thus,

$$\begin{aligned} N_{\kern 0.10emj\kern 0.10em}=N_{A}^{\left(\kern 0.10emj\kern 0.10em\right) }+N_{B}^{\left(\kern 0.10emj\kern 0.10em\right) }\sim \zeta _{+}^{\kern 0.10emj\kern 0.10em+2}. \end{aligned}$$

(45)

On the other hand, the fluctuations in the number of letters with respect to the asymptotic expectation values, gauged by

$$\begin{aligned} G_{j}=\left| N_{A}^{\left(\kern 0.10emj\kern 0.10em\right) }-x_{A}N_{\kern 0.10emj\kern 0.10em}\right| , \end{aligned}$$

(46)

are governed by

$$\begin{aligned} G_{j}\approx \frac{1}{\sqrt{4k+1}}\left| \zeta _{-}^{j}\left[ \zeta _{-}-x_{A}\left( \zeta _{-}-k\right) \right] \right| \propto \left| \zeta _{-}^{j}\right| \propto N_{j}^{\omega }, \end{aligned}$$

(47)

which defines the wandering exponent

$$\begin{aligned} \omega =\frac{\ln \left| \zeta _{-}\right| }{\ln \zeta _{+}}. \end{aligned}$$

(48)

If $\omega <0$, the geometrical fluctuations get smaller as the sequence gets larger, and at long times the behavior should recover that of the uniform limit. On the other hand, if $\omega >0$, fluctuations become larger and larger. The case $\omega =0$ is marginal and may give rise to nonuniversal behavior. For the generalized Fibonacci sequence, we have $\omega =-1<0$ for $k=1$, $\omega =0$ for $k=2$, and $\omega \approx 0.317>0$ for $k\ge 3$.

Appendix B. Diagonalizing the Matrix $\textbf{M}$

The matrix $\textbf{M}$ in Eq. (19) can be written as

$$\begin{aligned} \textbf{M}=\textbf{V}\left( \begin{array}{ccc} 1 &{} 0 &{} 0\\ 0 &{} \Xi _{-} &{} 0\\ 0 &{} 0 &{} \Xi _{+} \end{array}\right) \textbf{V}^{-1}, \end{aligned}$$

(49)

with $\Xi _{\pm }$ given by Eq. (23) and

$$\begin{aligned} \textbf{V}=\left( \begin{array}{ccc} -1 &{} -\zeta _{+}/k &{} -\zeta _{-}/k\\ 1 &{} \Xi _{+}/k^{2} &{} \Xi _{-}/k^{2}\\ 1 &{} 1 &{} 1 \end{array}\right) . \end{aligned}$$

(50)

Therefore,

$$\begin{aligned} \textbf{M}^{j}=\textbf{V}\left( \begin{array}{ccc} 1 &{} 0 &{} 0\\ 0 &{} \Xi _{-}^{\kern 0.10emj} &{} 0\\ 0 &{} 0 &{} \Xi _{+}^{\kern 0.10emj} \end{array}\right) \textbf{V}^{-1}. \end{aligned}$$

(51)

Using

$$\begin{aligned} \left( \begin{array}{c} \ln r_{0}^{++}\\ \ln r_{0}^{-}\\ \ln r_{0}^{--} \end{array}\right) =\left( \begin{array}{c} k\left( \mu -\lambda _{B}\right) \\ \mu -\lambda _{A}\\ \left( k+1\right) \left( \mu -\lambda _{A}\right) \end{array}\right) \end{aligned}$$

(52)

and

$$\begin{aligned} \left( \begin{array}{c} \Delta t_{0}^{++}\\ \Delta t_{0}^{-}\\ \Delta t_{0}^{--} \end{array}\right) =\left( \begin{array}{c} k\\ 1\\ k+1 \end{array}\right) \Delta t \end{aligned}$$

(53)

in Eqs. (17) and (18), we obtain Eqs. (21) and (22) with

$$\begin{aligned} \eta _{0}=\frac{\left( \mu -\lambda _{A}\right) -k\left( \mu -\lambda _{B}\right) }{k-2}=-\eta _{1}=-\eta _{2}, \end{aligned}$$

(54)

$$\begin{aligned} \eta _{0}^{\pm }=\mp \Delta \left[ \left( \Xi _{\pm }-k^{2}\right) \left( \mu -\lambda _{A}\right) +k\left( \zeta _{\pm }+k\zeta _{\mp }\right) \left( \mu -\lambda _{B}\right) \right] , \end{aligned}$$

(55)

$$\begin{aligned} \eta _{1}^{\pm }=\mp \Delta \left[ \left( \zeta _{\pm }+k\zeta _{\mp }\right) \left( \mu -\lambda _{A}\right) +k\left( \zeta _{\pm }-k\right) \left( \mu -\lambda _{B}\right) \right] , \end{aligned}$$

(56)

$$\begin{aligned} \eta _{2}^{\pm }= & {} \mp \Delta \left\{ \left[ \Xi _{\pm }-k\left( k-1\right) \zeta _{\pm }\right] \left( \mu -\lambda _{A}\right) \right. \nonumber \\{} & {} +\left. k\left( \Xi _{\pm }-k^{2}\right) \left( \mu -\lambda _{B}\right) \right\} , \end{aligned}$$

(57)

$$\begin{aligned} \tau _{0}=-\frac{k-1}{k-2}=-\tau _{1}=-\tau _{2}, \end{aligned}$$

(58)

$$\begin{aligned} \tau _{0}^{\pm }=\mp \Delta \left[ \left( \Xi _{\pm }-k^{2}\right) +k\left( \zeta _{\pm }+k\zeta _{\mp }\right) \right] , \end{aligned}$$

(59)

$$\begin{aligned} \tau _{1}^{\pm }=\mp \Delta \left( \zeta _{\pm }-k\left( k-1\right) \right) , \end{aligned}$$

(60)

$$\begin{aligned} \tau _{2}^{\pm }=\mp \Delta \left[ \left( k+1\right) \Xi _{\pm }-k\left( k-1\right) \zeta _{\pm }-k^{3}\right] , \end{aligned}$$

(61)

in which

$$\begin{aligned} \Delta ^{-1}=\left( k-2\right) \sqrt{1+4k}. \end{aligned}$$

(62)

It is interesting to notice that

$$\begin{aligned} \eta _{i}^{\pm }=\gamma _{i}^{\pm }\left( \mu -\frac{1}{\zeta _{\pm }}\lambda _{A}-\left( 1-\frac{1}{\zeta _{\pm }}\right) \lambda _{B}\right) , \end{aligned}$$

(63)

where $\gamma _{0}^{\pm }=\pm \Delta \left( \zeta _{\pm }\left( k^{2}-k-1\right) -k\right)$, $\gamma _{1}^{\pm }=\pm \Delta \left( k\left( k-1\right) -\zeta _{\pm }\right)$, and $\gamma _{2}^{\pm }=\pm \Delta \left( k\left( k^{2}-1\right) -\left( 2k+1\right) \zeta _{\pm }-k^{2}\zeta _{\mp }\right)$. It is easy to show that $\eta _{i}^{+}>0$ for $k\ge 0$.

For $k=2$, $\eta _{i}$, $\eta _{i}^{-}$, $\tau _{i}$ and $\tau _{i}^{-}$ are divergent. However, the following useful quantities remain finite:

$$\begin{aligned} \lim _{k\rightarrow 2}\left( \eta _{0}+\eta _{0}^{-}\right) =\frac{2}{9}\left( \mu +4\lambda _{A}-5\lambda _{B}\right) , \end{aligned}$$

(64)

$$\begin{aligned} \lim _{k\rightarrow 2}\left( \eta _{1}+\eta _{1}^{-}\right) =\frac{1}{9}\left( \mu -5\lambda _{A}-4\lambda _{B}\right) , \end{aligned}$$

(65)

$$\begin{aligned} \lim _{k\rightarrow 2}\left( \eta _{2}+\eta _{2}^{-}\right) =\frac{1}{9}\left( -5\mu -11\lambda _{A}+16\lambda _{B}\right) , \end{aligned}$$

(66)

$$\begin{aligned} \lim _{k\rightarrow 2}\eta _{0}^{+}=\frac{16}{9}\left[ \mu -\frac{1}{2}\left( \lambda _{A}+\lambda _{B}\right) \right] , \end{aligned}$$

(67)

$$\begin{aligned} \lim _{k\rightarrow 2}\eta _{1}^{+}=\frac{8}{9}\left[ \mu -\frac{1}{2}\left( \lambda _{A}+\lambda _{B}\right) \right] , \end{aligned}$$

(68)

$$\begin{aligned} \lim _{k\rightarrow 2}\eta _{2}^{+}=\frac{32}{9}\left[ \mu -\frac{1}{2}\left( \lambda _{A}+\lambda _{B}\right) \right] , \end{aligned}$$

(69)

$\lim _{k\rightarrow 2}\left( \tau _{0}+\tau _{0}^{-}\right) =\frac{2}{9},$ $\lim _{k\rightarrow 2}\left( \tau _{1}+\tau _{1}^{-}\right) =\frac{1}{9},$ $\lim _{k\rightarrow 2}\left( \tau _{2}+\tau _{2}^{-}\right) =-\frac{5}{9},$ $\lim _{k\rightarrow 2}\tau _{0}^{+}=\frac{16}{9},$ $\lim _{k\rightarrow 2}\tau _{1}^{+}=\frac{8}{9},$ and $\lim _{k\rightarrow 2}\tau _{2}^{+}=\frac{32}{9}$.