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A fixed-step isotropic random flight is a random walk in continuous space with independent steps of a common length $\ell$, each direction uniformly distributed over all directions. In $d$ dimensions, write its position as $S_n=\ell {\sum }_{j=1}^n{U}_j$, where the $U_j$ are independent unit vectors. Rotational symmetry gives $\mathbb{E}{U}_j=0$ and $\mathbb{E}(U_j{U}_j^{\mathsf{T}})=I_d/d$. Independence then givesThus 100 steps of length one have total path length 100 but root mean square displacement 10. A zero mean vector does not mean that a typical walker remains at its starting point.^[ref]

An exact two-step example in three dimensions shows how the endpoint spreads. The cosine $C=U_1\cdot U_2$ is uniform on $[-1,1]$, and $R^2=\Vert S_2{\Vert }^2=2{\ell }^2(1+C)$. Changing variables gives the radial probability density functionThis is a density per unit distance; a density per unit volume includes the spherical volume factor $4\pi r^2$. Uniform polar angle would give a different model: isotropic simulation in three dimensions samples the cosine of that angle uniformly.

In three dimensions the one-step characteristic function is $\sin (\ell k)/(\ell k)$, with value one at $k=0$, so the $n$-step characteristic function is its $n$th power. For a step duration $\tau$, the diffusion limit keeps $D={\ell }^2/(6\tau )$ fixed as $\ell ,\tau$ tend to zero. It yields the Gaussian diffusion equation kernel. At finite $n$, however, the walker cannot travel farther than $n\ell$; a Gaussian distribution is a limiting approximation, not the exact finite-step law.^[ref]

For conservative photon transport at constant speed $v$, suppose successive scattering directions are independent and isotropically distributed, and free-path lengths follow an exponential distribution of mean $\ell$. In three spatial dimensions, the resulting long-time diffusion coefficient isHere $\ell$ is the scattering mean free path. These assumptions describe homogeneous, memoryless interactions without absorption.

The velocity autocorrelation function gives a direct derivation. An initially isotropic ensemble has zero mean velocity. During a lag $s$, the probability of no scattering is $e^{-vs/\ell }$. No scattering preserves the direction; one or more independent isotropic scattering events give zero expected directional correlation. ConsequentlyThe three-dimensional Green-Kubo formula for diffusion now yieldsThe factor $1/3$ converts the dot-product correlation into one Cartesian component under isotropy.

The same correlation gives the full mean squared displacement:At short times this is $v^2{t}^2$; at long times its leading term is $6Dt$. The model therefore connects direct flights to the diffusive many-scattering limit.

The free-path distribution matters. A fixed-step isotropic random flight with every length equal to $\ell$ has mean-square displacement $n{\ell }^2$ after $n$ steps and elapsed time $n\ell /v$, giving $D=v\ell /6$. Replacing exponential flights by fixed-length flights thus changes the coefficient by a factor of two. For anisotropic scattering, directional persistence instead leads to a transport mean free path; strong absorption and boundaries require additional modeling.

V. Balakrishnan is a theoretical physicist associated with the Indian Institute of Technology Madras. Its institutional profile identifies research in quantum dynamics, stochastic processes and dynamical systems, including random walks on hierarchical structures and recurrence statistics.^[ref]

His Physical Applications of Stochastic Processes by V. Balakrishnan connects probability calculations to physical motion. A particularly instructive example is the fixed-step isotropic random flight: independent directions give zero mean displacement but a mean squared displacement proportional to the number of steps. Shrinking the step length and duration together then produces a finite diffusion coefficient and a diffusion equation description.^[ref]

The example illustrates a recurring scientific distinction in his exposition: an average position, a typical displacement and a probability density answer different questions. Cancellation of displacement vectors does not eliminate the spreading of individual trajectories. IIT Madras's account of the chair established in his honor also identifies contributions to continuous-time random walks and recurrences in dynamical systems.^[ref]