Model stochastic transitions with clear rate matrices. Choose Euler or RK4 integration for stability better. View probabilities instantly, then download tables and summaries today.
Three-state cyclic kinetics example (rates in 1/time).
| Rate | Value | Rate | Value |
|---|---|---|---|
| q12 | 1.2 | q13 | 0.4 |
| q21 | 0.6 | q23 | 0.9 |
| q31 | 0.3 | q32 | 1.1 |
| p(0) | [1, 0, 0] | t_end | 10 |
Use these values to see relaxation toward a steady distribution.
For states i=1..N, the probability row-vector p(t) evolves as:
Off-diagonal qij are transition rates from i to j. Diagonal terms conserve total probability:
Each row sums to zero, so Σ p(t) stays one.
A master equation describes how probabilities move between discrete states under random transitions. This calculator solves continuous-time Markov dynamics by constructing a generator matrix Q from your rates and integrating p(t) forward in time. It is suited for kinetic networks, switching systems, and reliability models where outcomes are best summarized as probabilities rather than single trajectories.
Master equations appear in chemical reaction kinetics (isomerization and catalytic cycles), charge trapping in semiconductors, population transfer in optical pumping, radiation damage and defect annealing, and coarse-grained diffusion on lattices. In many labs, the “state” represents a measurable configuration, such as occupancy, conformer, or device mode.
You enter off-diagonal rates qij (i → j) with units of 1/time. The solver computes each diagonal term as qii = −∑j≠i qij, ensuring each row sums to zero. That constraint conserves total probability in exact arithmetic and matches physical bookkeeping: probability leaving state i must reappear elsewhere.
The output table shows p(t) for every state at sampled times. Rapid changes indicate fast channels or strong asymmetry in rates, while slow drifts reflect bottlenecks. If your initial distribution is localized, you can quantify mixing by observing how quickly probabilities spread across the network. Exported CSV files support plotting and parameter sweeps.
Euler is simple and fast, but may require small dt for stability, especially when the largest rate is high. RK4 uses four slope evaluations per step and usually allows larger dt while maintaining accuracy. For stiff systems, use RK4 and reduce dt until results stop changing materially in the final distribution.
A useful heuristic is dt × max(outgoing rate) ≪ 1. For example, if a state has total outgoing rate 50 s⁻¹, choosing dt = 0.001 s gives dt·rate = 0.05, which is typically safe. If dt is too large, Euler can produce negative probabilities; the optional clipping and renormalization controls provide robust, report-ready outputs.
Many networks converge to a stationary distribution π satisfying πQ = 0 with ∑π = 1. The solver estimates this steady state using a constrained linear solve. Compare π to p(t_end): close agreement suggests your simulation time is long enough. If detailed balance holds, steady state may align with equilibrium expectations from thermodynamic arguments.
The PDF summary captures inputs, p(0), p(t_end), the steady-state estimate, and the computed Q matrix. This makes it easy to document models in publications, lab notebooks, or design reviews. When you tune rates against data, keep versioned CSV exports so parameter changes and fit outcomes remain traceable and defensible.
A rate qij is the transition intensity from state i to state j, in 1/time. Larger values mean faster probability flow along that edge.
The diagonal enforces conservation: qii equals the negative sum of outgoing rates from i. This guarantees each row of Q sums to zero.
Reduce dt, switch to RK4, and enable clipping or renormalization. Negatives often indicate an unstable step size, not a physical effect.
Increase t_end and see whether p(t_end) changes. If it stabilizes and matches the steady-state estimate, your run likely reached long-time behavior.
Yes. Set all outgoing rates from an absorbing state to zero. The computed diagonal becomes zero, and probability can accumulate there over time.
The CSV includes time and state probabilities for the full trajectory. The PDF summarizes inputs, p(0), p(t_end), the steady-state estimate, and the computed Q matrix.
This version supports 2 to 6 states for usability and performance. If you need larger networks, the same structure can be extended with file upload and sparse methods.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.