Current Fluctuations in One-Dimensional Diffusion-Reaction Systems via Tensor Networks

TL;DR

The paper develops a tensor-network framework to compute the cumulant generating function $Q(\lambda)$ of current statistics in a 1D diffusion-reaction system with holes and electrons in a semiconducting material. It represents the state distribution as a matrix product state (MPS) and the tilted generator as a matrix product operator (MPO), and uses DMRG to extract the leading eigenvalue that yields $Q(\lambda)$. The results confirm Gallavotti–Cohen symmetry and show that activating the reaction between carriers damps current fluctuations, establishing an upper bound $\tfrac{2D}{J} \le \coth(A/2)$ under symmetric boundary conditions, with equality in the purely diffusive limit. The study demonstrates the power of tensor-network methods for nonequilibrium classical stochastic systems and outlines natural extensions to more complex geometries and rate laws.

Abstract

Tensor networks are employed to characterize the current fluctuations in one-dimensional diffusion-reaction systems. The representative system under study is a semiconducting material where holes and electrons constitute two types of charge carriers. These holes and electrons diffuse in the system with the reactions of pair-generation and -recombination occurring between them. The system is driven by imbalanced conditions imposed at two boundaries. The large deviation function encoding the full counting statistics of electric current is numerically calculated using the density matrix renormalization group. The fluctuation theorem is shown to hold for the current. Moreover, by comparing the cases where the reactions are turned on or off, it is revealed that the reactions have a damping effect on current fluctuations. This indicates an interesting inequality, suggesting that current fluctuations are upper bounded.

Figures (6)

• Figure 1: Schematic diagram of a 1D semiconducting material. The white dots represent holes and the black ones represent electrons. The system is in contact at the ends with two reservoirs that fix the densities of holes and electrons.
• Figure 2: Graphical notations of the distribution function $F_{\bf PN}$ represented as an MPS (left panel) and the tilted generator $\hat{L}_{\lambda{\bf P'N'}}^{\bf PN}$ represented as an MPO (right panel). The composite indices $\{P_iN_i\}$ are used to describe the state of each discretized cell. Only five sites are shown for illustration. In the MPO, the counting parameter $\lambda$ is included in the first site.
• Figure 3: Grayscale map of the steady-state probability distribution of holes and electrons $\{P_3,N_3\}$ in the third cell. Five contours and a gray bar indicating the probability values are shown. The parameter values listed in Table \ref{['tab_values']} plus $k_+=k_-=10.0$ are used in tensor-network calculations. The annotation is the Pearson correlation coefficient between the probability distributions of $P_3$ and $N_3$.
• Figure 4: The profiles of the average occupation number of holes $\langle P\rangle$ along the system. The lines are results from tensor-network calculations, while the asterisks from solving the ODEs (\ref{['eq_ode1']})-(\ref{['eq_ode4']}). The steplike shape of the lines originates from the discretization in space. The boundary conditions used for solving the ODEs are $p(-0.05)=\bar{p}_{\rm L}$, $n(-0.05)=\bar{n}_{\rm L}$, $p(1.05)=\bar{p}_{\rm R}$, and $n(1.05)=\bar{n}_{\rm R}$. The purpose of shifting the boundaries is to obtain consistent results with those from tensor-network calculations in the discretized scheme. Two cases where the reactions are turned on ($k_+=k_-=10.0$) and off ($k_+=k_-=0$) are shown for comparison. The inset shows the deviation between the two profiles. Other parameter values are given in Table \ref{['tab_values']}.
• Figure 5: Convergence of the cumulant generating function calculated with DMRG as the the number of sweeps is increased. The reactions are turned off, $k_+=k_-=0$, and in this case, analytical solution can be obtained, shown as the dashed line in the inset. The dots represent results calculated using the DMRG algorithm. Different colors correspond to the cases with different numbers of sweeps. The asterisks with solid line joining them represent the errors that are root mean square of the differences between the results calculated using the DMRG algorithm and the analytical solution. The parameter values in Table \ref{['tab_values']} are taken. Prior to DMRG calculations, the system is initialized to the steady state such that the number of holes or electrons is Poisson distributed in each cell with the mean value determined from the linear profile shown as the blue line in Fig \ref{['fig_profile']}. In DMRG calculations with a high-performance laptop, it took about 380 s with 100 sweeps to compute one data point of $Q(\lambda)$. This time cost may vary depending on the computing platform and the detailed DMRG implementation.