Brownian Motion¶

Brownian motion describes motion of small particles with stochastic forces applied to them. The math of Brownian motion can be modeled with Wiener process. In this tutorial, we take a simple form of the model and treat the stochastic forces as Gaussian.

For consistency, we always use $\mathbf x$ for displacement, and $t$ for steps. The model we are using is

$$ \begin{align} \mathbf x(t + \mathrm dt) &= \mathbf x(t) + \mathcal{N}(\mu=0, \sigma=\sigma \sqrt{\mathrm d t}) \end{align} $$

Read more:

1. Brownian motion and random walks. [cited 13 Mar 2024]. Available: https://web.mit.edu/8.334/www/grades/projects/projects17/OscarMickelin/brownian.html
2. Contributors to Wikimedia projects. Brownian motion. In: Wikipedia [Internet]. 22 Jan 2024 [cited 13 Mar 2024]. Available: https://en.wikipedia.org/wiki/Brownian_motion

hamilflow implemented a Brownian motion model called BrownianMotion.

In [1]:

import plotly.express as px

from hamilflow.models.brownian_motion import BrownianMotion

import plotly.express as px from hamilflow.models.brownian_motion import BrownianMotion

1D Brownian Motion¶

In [2]:

bm_1d = BrownianMotion(
    system={
        "sigma": 1,
        "delta_t": 1,
    },
    initial_condition={"x0": 0},
)

bm_1d = BrownianMotion( system={ "sigma": 1, "delta_t": 1, }, initial_condition={"x0": 0}, )

Call the model to generate 1000 steps.

In [3]:

df_1d = bm_1d.generate_from(n_steps=1000)

df_1d = bm_1d.generate_from(n_steps=1000)

In [4]:

px.line(df_1d, x="t", y="x_0")

px.line(df_1d, x="t", y="x_0")

2D Brownian Motion¶

Our BrownianMotion model calculates the dimension of the space based on the dimension of the initial condition $x_0$. To create a 2D Brownian motion model, we need the initial condition to be length 2.

In [5]:

bm_2d = BrownianMotion(
    system={
        "sigma": 1,
        "delta_t": 1,
    },
    initial_condition={"x0": [0, 0]},
)

bm_2d = BrownianMotion( system={ "sigma": 1, "delta_t": 1, }, initial_condition={"x0": [0, 0]}, )

We call the model to generate 1000 steps.

In [6]:

df_2d = bm_2d.generate_from(n_steps=500)

df_2d = bm_2d.generate_from(n_steps=500)

In [7]:

(
    px.scatter(df_2d, x="x_0", y="x_1", color="t")
    .update_traces(
        mode="lines+markers",
        marker={
            "size": 2.5,
        },
        line={"width": 1},
    )
    .update_yaxes(
        scaleanchor="x",
        scaleratio=1,
    )
)

( px.scatter(df_2d, x="x_0", y="x_1", color="t") .update_traces( mode="lines+markers", marker={ "size": 2.5, }, line={"width": 1}, ) .update_yaxes( scaleanchor="x", scaleratio=1, ) )

In [ ]: