Mean First Passage Time of the Symmetric Noisy Voter Model with Arbitrary Initial and Boundary Conditions

TL;DR

This work derives exact analytical expressions for the mean first passage time in the symmetric noisy voter model under arbitrarily placed absorbing and reflective boundaries. By solving the backward Fokker–Planck equation, the authors obtain a general MFPT solution in terms of the incomplete beta function and the Meijer G-function, and they present explicit formulas for two-boundary problems. They show that MFPT is inherently asymmetric with respect to the initial condition when boundaries are not symmetric, and that this asymmetry is magnified by independent-transition rate ε; in the large-ε limit, MFPT obeys a Kramers-like law, scaling exponentially with ε. Special ε values (ε = n/2) yield simplified, elementary expressions, including arctangent and arcsin forms, enabling clear interpretation of symmetry conditions. The results are validated against numerical simulations and offer applicable insights for threshold-driven social systems and diffusion-like physical processes.

Abstract

Models of imitation and herding behavior often underestimate the role of individualistic actions and assume symmetric boundary conditions. However, real-world systems (e.g., electoral processes) frequently involve asymmetric boundaries. In this study, we explore how arbitrarily placed boundary conditions influence the mean first passage time in the symmetric noisy voter model, and how individualistic behavior amplifies this asymmetry. We derive exact analytical expressions for mean first passage time that accommodate any initial condition and two types of boundary configurations: (i) both boundaries absorbing, and (ii) one absorbing and one reflective. In both scenarios, mean first passage time exhibits a clear asymmetry with respect to the initial condition, shaped by the boundary placement and the rate of independent transitions. Symmetry in mean first passage time emerges only when absorbing boundaries are equidistant from the midpoint. Additionally, we show that Kramers' law holds in both configurations when the rate of independent transitions is large. Our analytical results are in excellent agreement with numerical simulations, reinforcing the robustness of our findings.

Figures (5)

• Figure 1: The mean first passage time $\overline{T}$ dependence on the initial condition $x_{0}$ with various values of independent transition rate $\varepsilon$ for the case with absorbing boundary conditions at $L$ and $H$. The black curves correspond to Eq. \ref{['eq:mfpt-lh-aa']}, while the red squares represent estimates obtained by numerical simulation of Eq. \ref{['eq:sdex']} with $h=1$ and $\varepsilon_{0}=\varepsilon_{1}=\varepsilon$. Boundary conditions were placed at $L=0.03$ and $H=0.63$.
• Figure 2: The mean first passage time $\overline{T}$ dependence on the initial condition $x_{0}$ with various values of independent transition rate $\varepsilon$ for the case with reflective boundary condition at $L$ and absorbing boundary condition at $H$. The black curves correspond to Eq. \ref{['eq:mfpt-lh-ra']}, while the red squares represent estimates obtained by numerical simulation of Eq. \ref{['eq:sdex']} with $h=1$ and $\varepsilon_{0}=\varepsilon_{1}=\varepsilon$. In all instances boundary conditions were placed at $L=0.03$ and $H=0.63$.
• Figure 3: The mean first passage time dependence $\overline{T}$ on the initial condition $x_{0}$ with symmetric absorbing boundaries (a), and with asymmetric absorbing boundaries (b). Colored curves show the MFPT dependence obtained by numerical simulation with different rates: $\varepsilon=0$ (red curves), $0.8$ (green), $1.6$ (blue), $2.4$ (cyan), $3.2$ (yellow), $4.0$ (magenta). Boundary conditions for the symmetric case (subfigure (a)) were placed at $L=0.2$ and $H=0.8$, for the asymmetric case (subfigure (b)) they were placed at $L=0.1$ and $H=0.7$.
• Figure 4: Numerically simulated dependence of $x_{0,\text{max}}$ on the placement of the asymmetric absorbing boundary conditions. Lower boundary is fixed at $L=0.05$, while the higher boundary $H$ is treated as an independent variable. Dashed black line highlights the midpoint between the boundary conditions, $\tfrac{H+L}{2}$. Colored symbols show numerical simulation results obtained with different rates: $\varepsilon=0$ (red squares), $1.6$ (green circles), $3.2$ (blue diamonds).
• Figure 5: The mean first passage time $\overline{T}$ dependence on the independent transition rate $\varepsilon$: (a) the case with absorbing boundary conditions at $L$ and $H$, (b) the case with reflective boundary condition at $L$ and absorbing boundary condition at $H$. The black curves correspond to Eq. \ref{['eq:mfpt-lh-aa']} (for subfigure (a)) and Eq. \ref{['eq:mfpt-lh-ra']} (for subfigure (b)), the gray curves follow Kramers' law, Eq. \ref{['eq:kramers-law']}. The red squares represent estimates obtained by numerical simulation of Eq. \ref{['eq:sdex']} with $h=1$ and $\varepsilon_{0}=\varepsilon_{1}=\varepsilon$. The insets show a more detailed dependence in the range of smaller $\varepsilon$. The boundary and initial conditions were set as follows $L=0.03$, $H=0.63$ and $x_{0}=0.23$.