Random Walk Variance Calculator

This calculator supports two common variance models. Choose the one that matches your experiment.

• Discrete steps (±a): Each step is +a with probability p and −a with probability q = 1 − p. For N independent steps, the mean per axis is ⟨X⟩ = N a (2p − 1), and the variance per axis is Var(X) = 4 N a² p(1 − p).
• Diffusion (Brownian): In one axis, the variance grows linearly: Var(X) = 2 D t. In d dimensions with independent axes, the mean-squared displacement is ⟨r²⟩ = 2 d D t.

Notes: For the discrete model, the unbiased case is p = 0.5, giving Var(X) = N a².

1. Select a model that matches your random motion.
2. Enter dimensions, then fill the model inputs.
3. Press Calculate to see results above.
4. Use CSV for spreadsheets and PDF for sharing.
5. Enable simulation to sanity-check theoretical values.

Random Walk Variance Guide

1) Purpose and scope

Random walks describe stepwise motion in gases, charge transport, price ticks, and molecular diffusion. This calculator focuses on variance, because variance is the cleanest measure of spread and uncertainty. It reports per-axis variance, standard deviation, and total mean-squared displacement in d dimensions.

2) What variance tells you

Variance quantifies how far typical outcomes deviate from the mean. If the mean is near zero, variance controls the “width” of the displacement distribution. The square root of variance, the standard deviation, gives a direct distance scale for expected fluctuations.

3) Discrete steps model

In the discrete option, each step is +a with probability p and −a with probability 1−p. After N independent steps, variance grows linearly with N: Var(X)=4Na²p(1−p). For the unbiased case p=0.5, this reduces to Var(X)=Na².

4) Drift and bias effects

When p≠0.5, the walk has drift. The mean displacement becomes ⟨X⟩=Na(2p−1), while variance still scales with N. In practice, drift can dominate the total spread, so the calculator also reports ⟨r²⟩=d(Var(X)+⟨X⟩²) to combine random scatter with systematic motion.

5) Dimensional scaling

Many experiments are naturally two- or three-dimensional. With independent and identical axes, total mean-squared displacement scales as d. That means doubling dimensions doubles ⟨r²⟩ under the same per-axis statistics, which is useful for comparing planar and volumetric motion.

6) Diffusion (continuous-time) limit

In the diffusion mode, the model assumes Brownian motion with diffusion coefficient D. The per-axis variance is Var(X)=2Dt, and the total mean-squared displacement is ⟨r²⟩=2dDt. These relations are standard in transport theory and connect macroscopic spread to microscopic agitation.

7) Simulation as a consistency check

The optional Monte Carlo block generates sample paths and estimates r². Agreement between the simulated E[r²] and the theoretical value builds confidence in your parameter choices. If you increase M, the sample mean stabilizes and random estimation noise shrinks.

8) Practical interpretation

Use variance to set error bars, choose sampling windows, and compare competing models. In tracking problems, variance helps decide whether observed spreading is consistent with unbiased diffusion or indicates directional forcing. For design, it provides a simple scaling law: spread grows with step count or time, and with the square of step size.

FAQs

1) What does “per-axis variance” mean?

It is the variance of displacement along one coordinate direction. For independent axes, total mean-squared displacement equals the sum of per-axis second moments.

2) Why can ⟨r²⟩ be larger than d·Var(X)?

When drift exists, the mean displacement is nonzero. Then ⟨r²⟩ includes both randomness and systematic motion through Var(X)+⟨X⟩² on each axis.

3) What inputs create an unbiased walk?

Set probability p to 0.5 in the discrete model. That makes the mean displacement zero, and the spread is governed purely by variance growth with N.

4) How do I choose step size a?

Use the physical step length per update in your experiment or simulation. If you change the time step, adjust a accordingly so the modeled displacement matches your sampling resolution.

5) When should I use the diffusion option?

Use diffusion when motion is well-approximated as continuous-time Brownian behavior and you know D. It is especially suitable for long-time, many-collision regimes.

6) Why might the Monte Carlo check be skipped?

Large values of M, d, and N can be slow in a single web request. Reduce one of them to keep computation responsive, or run heavy simulations offline.

7) What is the main output to report in papers?

Often report Var(X) per axis with units, plus ⟨r²⟩ in d dimensions. Together they communicate both coordinate-level spreading and overall displacement scale.