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Rotational diffusion

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Rotational diffusion is the rotational movement which acts upon any object such as <a href="Particle" title="Particle">particles</a>, <a href="Molecule" title="Molecule">molecules</a>, <a href="Atom" title="Atom">atoms</a> when present in a <a href="Fluid" title="Fluid">fluid</a>, by random changes in their <a href="Orientation_(geometry)" title="Orientation (geometry)">orientations</a>.

Although the directions and intensities of these changes are <a href="Statistics" title="Statistics">statistically</a> random, they do not arise randomly and are instead the result of interactions between particles. One example occurs in <a href="Colloid" title="Colloid">colloids</a>, where relatively large <a href="Solubility" title="Solubility">insoluble</a> particles are suspended in a greater amount of fluid. The changes in orientation occur from <a href="Collision" title="Collision">collisions</a> between the particle and the many molecules forming the fluid surrounding the particle, which each transfer <a href="Kinetic_energy" title="Kinetic energy">kinetic energy</a> to the particle, and as such can be considered <a href="Randomness" title="Randomness">random</a> due to the varied <a href="Speed" title="Speed">speeds</a> and amounts of fluid molecules incident on each individual particle at any given time.

The analogue to <a href="Diffusion" title="Diffusion">translational diffusion</a> which determines the particle's position in <a href="Space" title="Space">space</a>, rotational diffusion randomises the orientation of any <a href="Particle" title="Particle">particle</a> it acts on.

Anything in a solution will experience rotational diffusion, from the <a href="Microscopic_scale" title="Microscopic scale">microscopic scale</a> where individual atoms may have an effect on each other, to the <a href="Macroscopic_scale" title="Macroscopic scale">macroscopic scale</a>.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Rotational diffusion has multiple applications in chemistry and physics, and is heavily involved in many biology based fields. For example, <a href="Protein%E2%80%93protein_interaction" title="Protein–protein interaction">protein-protein interaction</a> is a vital step in the communication of biological signals. In order to communicate, the proteins must both come into contact with each other and be facing the appropriate way to interact with each other's <a href="Binding_site" title="Binding site">binding site</a>, which relies on the proteins ability to rotate.<a href="#cite_note-1">[1]</a>

As an example concerning physics, <a href="Rotational_Brownian_motion_(astronomy)" title="Rotational Brownian motion (astronomy)">rotational Brownian motion in astronomy</a> can be used to explain the orientations of the orbital planes of <a href="Binary_star" title="Binary star">binary stars</a>, as well as the seemingly random spin axes of <a href="Supermassive_black_hole" title="Supermassive black hole">supermassive black holes</a>.<a href="#cite_note-DM-2">[2]</a>

The random re-orientation of molecules (or larger systems) is an important process for many <a href="Biophysics" title="Biophysics">biophysical</a> probes. Due to the <a href="Equipartition_theorem" title="Equipartition theorem">equipartition theorem</a>, larger molecules re-orient more slowly than do smaller objects and, hence, measurements of the rotational <a href="Diffusion_equation" title="Diffusion equation">diffusion constants</a> can give insight into the overall mass and its distribution within an object. Quantitatively, the mean square of the <a href="Angular_velocity" title="Angular velocity">angular velocity</a> about each of an object's <a href="Principal_axis_(mechanics)" class="mw-redirect" title="Principal axis (mechanics)">principal axes</a> is inversely proportional to its <a href="Moment_of_inertia" title="Moment of inertia">moment of inertia</a> about that axis. Therefore, there should be three rotational diffusion constants - the eigenvalues of the rotational diffusion tensor - resulting in five rotational <a href="Time_constant" title="Time constant">time constants</a>.<a href="#cite_note-perrin_1934-3">[3]</a><a href="#cite_note-perrin_1936-4">[4]</a> If two eigenvalues of the diffusion tensor are equal, the particle diffuses as a <a href="Spheroid" title="Spheroid">spheroid</a> with two unique diffusion rates and three time constants. And if all eigenvalues are the same, the particle diffuses as a <a href="Sphere" title="Sphere">sphere</a> with one time constant. The diffusion tensor may be determined from the <a href="Perrin_friction_factors" title="Perrin friction factors">Perrin friction factors</a>, in analogy with the <a href="Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation</a> of translational diffusion, but often is inaccurate and direct measurement is required.

The rotational diffusion tensor may be determined experimentally through <a href="Fluorescence_anisotropy" title="Fluorescence anisotropy">fluorescence anisotropy</a>, <a href="Flow_birefringence" title="Flow birefringence">flow birefringence</a>, <a href="Dielectric_spectroscopy" title="Dielectric spectroscopy">dielectric spectroscopy</a>, <a href="Relaxation_(NMR)" title="Relaxation (NMR)">NMR relaxation</a> and other biophysical methods sensitive to picosecond or slower rotational processes. In some techniques such as fluorescence it may be very difficult to characterize the full diffusion tensor, for example measuring two diffusion rates can sometimes be possible when there is a great difference between them, e.g., for very long, thin ellipsoids such as certain <a href="Virus" title="Virus">viruses</a>. This is however not the case of the extremely sensitive, atomic resolution technique of NMR relaxation that can be used to fully determine the rotational diffusion tensor to very high precision. Rotational diffusion of macromolecules in complex biological fluids (i.e., cytoplasm) is slow enough to be measurable by techniques with microsecond time resolution, i.e. <a href="Fluorescence_correlation_spectroscopy" title="Fluorescence correlation spectroscopy">fluorescence correlation spectroscopy</a>.<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="The_diffusion_equation_and_the_rotational_diffusion_constant">The diffusion equation and the rotational diffusion constant</h2></div>

To model the diffusion process, consider a large number of identical rotating particles.

The orientation of each particle is described by a <a href="Unit_vector" title="Unit vector">unit vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation>

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</math><img src="./d125dccc556f5c8b0bf98a4f3847590b3f353bd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}}" loading="lazy">; for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation>

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</math><img src="./d125dccc556f5c8b0bf98a4f3847590b3f353bd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}}" loading="lazy"> might represent the orientation of an <a href="Electrical_dipole_moment" class="mw-redirect" title="Electrical dipole moment">electric</a> or <a href="Magnetic_dipole_moment" class="mw-redirect" title="Magnetic dipole moment">magnetic dipole moment</a>. Let f(θ, φ, t) represent the <a href="Probability_density_function" title="Probability density function">probability density distribution</a> for the orientation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation>

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</math><img src="./d125dccc556f5c8b0bf98a4f3847590b3f353bd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}}" loading="lazy"> at time t. Here, θ and φ represent the <a href="Spherical_coordinate_system" title="Spherical coordinate system">spherical angles</a>, with θ being the polar angle between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation>

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</math><img src="./d125dccc556f5c8b0bf98a4f3847590b3f353bd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}}" loading="lazy"> and the z-axis and φ being the <a href="Azimuth" title="Azimuth">azimuthal angle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation>

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</math><img src="./d125dccc556f5c8b0bf98a4f3847590b3f353bd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}}" loading="lazy"> in the x-y plane.

<a href="Fick's_laws_of_diffusion#Fick's_second_law" title="Fick's laws of diffusion">Fick's second law of diffusion</a>, applied to angular diffusion, states that in the absence of an external torque on the particles, the evolution of f(θ, φ, t) obeys

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{D_{\mathrm {rot} }}}{\frac {\partial f}{\partial t}}=\nabla _{\theta \phi }^{2}f.}</annotation>

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</math><img src="./cec00889a0805669f4ff55a762020757f5660c67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.474ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{D_{\mathrm {rot} }}}{\frac {\partial f}{\partial t}}=\nabla _{\theta \phi }^{2}f.}" loading="lazy"></dd></dl>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mathrm {rot} }}">

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<annotation encoding="application/x-tex">{\displaystyle D_{\mathrm {rot} }}</annotation>

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</math><img src="./828330e8741635575af0a9b51f97b097dc85e220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.263ex; height:2.509ex;" alt="{\displaystyle D_{\mathrm {rot} }}" loading="lazy"> is the angular diffusion coefficient, whose units are rad2/s.

This equation contains the <a href="Laplace_operator" title="Laplace operator">angular Laplace operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{\theta \phi }^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \nabla _{\theta \phi }^{2}}</annotation>

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</math><img src="./f4bc9334164eeb84b3540cbd4cca2ec06d89fdaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.919ex; height:3.509ex;" alt="{\displaystyle \nabla _{\theta \phi }^{2}}" loading="lazy">, which can be written

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<annotation encoding="application/x-tex">{\displaystyle \nabla _{\theta \phi }^{2}f={\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}.}</annotation>

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</math><img src="./f22ac0ffa5819115f1b1def98a2579bb474aa8fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.589ex; height:6.343ex;" alt="{\displaystyle \nabla _{\theta \phi }^{2}f={\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Solution_of_the_diffusion_equation">Solution of the diffusion equation</h3></div>

This <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a> may be solved using the method of <a href="Separation_of_variables" title="Separation of variables">separation of variables</a> by expanding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta ,\phi ,t)}">

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<annotation encoding="application/x-tex">{\displaystyle f(\theta ,\phi ,t)}</annotation>

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</math><img src="./24b30ccb00699daca773ddc6312daeedb1736ed1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.471ex; height:2.843ex;" alt="{\displaystyle f(\theta ,\phi ,t)}" loading="lazy"> in <a href="Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{l}^{m}(\theta ,\phi ).}">

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<annotation encoding="application/x-tex">{\displaystyle Y_{l}^{m}(\theta ,\phi ).}</annotation>

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</math><img src="./396699f97189fd89b2387af7bd67f0cb5b6fabd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.541ex; height:3.009ex;" alt="{\displaystyle Y_{l}^{m}(\theta ,\phi ).}" loading="lazy">

Since spherical harmonics satisfy the identity

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<annotation encoding="application/x-tex">{\displaystyle \nabla _{\theta \phi }^{2}Y_{l}^{m}(\theta ,\phi )=-l(l+1)Y_{l}^{m}(\theta ,\phi ),}</annotation>

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</math><img src="./d61eccdb9b6f7fadcbc4fc8a24cc83085fc544ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:34.46ex; height:3.509ex;" alt="{\displaystyle \nabla _{\theta \phi }^{2}Y_{l}^{m}(\theta ,\phi )=-l(l+1)Y_{l}^{m}(\theta ,\phi ),}" loading="lazy">.</dd></dl>

the solution may be written

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<annotation encoding="application/x-tex">{\displaystyle f(\theta ,\phi ,t)=\sum _{l=0}^{\infty }\sum _{m=-l}^{l}C_{lm}Y_{l}^{m}(\theta ,\phi )e^{-t/\tau _{l}}}</annotation>

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</math><img src="./6bc8d6e377517dd79da5cf555c71a6d0ff5d622f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.201ex; height:7.509ex;" alt="{\displaystyle f(\theta ,\phi ,t)=\sum _{l=0}^{\infty }\sum _{m=-l}^{l}C_{lm}Y_{l}^{m}(\theta ,\phi )e^{-t/\tau _{l}}}" loading="lazy">,</dd></dl>

where Clm are constants (which depend on the initial distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta ,\phi ,0)}">

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<annotation encoding="application/x-tex">{\displaystyle f(\theta ,\phi ,0)}</annotation>

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</math><img src="./5fa7250f5988373f92509b9b4cc23dcf2beb185d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.794ex; height:2.843ex;" alt="{\displaystyle f(\theta ,\phi ,0)}" loading="lazy">) and the time constants are

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<annotation encoding="application/x-tex">{\displaystyle \tau _{l}={\frac {1}{D_{\mathrm {rot} }l(l+1)}}}</annotation>

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</math><img src="./f41f6396c8f2dd9b05cb654087243265e70be8ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.135ex; height:6.009ex;" alt="{\displaystyle \tau _{l}={\frac {1}{D_{\mathrm {rot} }l(l+1)}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Two-dimensional_rotational_diffusion">Two-dimensional rotational diffusion</h2></div>

A sphere rotating around a fixed axis will rotate in two <a href="Dimension" title="Dimension">dimensions</a> only and can be viewed from above the fixed axis as a circle. In this example, a sphere which is fixed on the vertical axis rotates around that axis only, meaning that the particle can have a θ value of 0 through 360 degrees, or 2π Radians, before having a net rotation of 0 again.<a href="#cite_note-8">[7]</a>

These directions can be placed onto a graph which covers the entirety of the possible positions for the <a href="Face_(geometry)" title="Face (geometry)">face</a> to be at relative to the starting point, through 2π radians, starting with -π radians through 0 to π radians. Assuming all particles begin with single orientation of 0, the first measurement of directions taken will resemble a <a href="Dirac_delta_function" title="Dirac delta function">delta function</a> at 0 as all particles will be at their starting, or 0th, position and therefore create an infinitely steep single line. Over time, the increasing amount of measurements taken will cause a spread in results; the initial measurements will see a thin peak form on the graph as the particle can only move slightly in a short time. Then as more time passes, the chance for the molecule to rotate further from its starting point increases which widens the peak, until enough time has passed that the measurements will be evenly distributed across all possible directions.

The distribution of orientations will reach a point where they become <a href="Continuous_uniform_distribution" title="Continuous uniform distribution">uniform</a> as they all randomly <a href="Statistical_dispersion" title="Statistical dispersion">disperse</a> to be nearly equal in all directions. This can be visualized in two ways.

<ol><li>For a single particle with multiple measurements taken over time. A particle which has an area designated as its face pointing in the starting orientation, starting at a time t0 will begin with an orientation distribution resembling a single line as it is the only measurement. Each successive measurement at time greater than t0 will widen the peak as the particle will have had more time to rotate away from the starting position.</li>

<li>For multiple particles measured once long after the first measurement. The same case can be made with a large number of molecules, all starting at their respective 0th orientation. Assuming enough time has passed to be much greater than t0, the molecules may have fully rotated if the forces acting on them require, and a single measurement shows they are near-to-evenly distributed.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Basic_equations">Basic equations</h3></div>

For rotational diffusion about a single axis, the mean-square angular deviation in time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle \left\langle \theta ^{2}\right\rangle =2D_{r}t}</annotation>

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</math><img src="./b0040915616372867dc104a3bba798db20ae9008.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.338ex; height:3.343ex;" alt="{\displaystyle \left\langle \theta ^{2}\right\rangle =2D_{r}t}" loading="lazy">,</dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{r}}">

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</math><img src="./8783bccae1e365d5e58cd502bea46ce4eee7fe34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.898ex; height:2.509ex;" alt="{\displaystyle D_{r}}" loading="lazy"> is the rotational diffusion coefficient (whose units are radians2/s).

The angular drift velocity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{d}=(d\theta /dt)_{\rm {drift}}}">

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<annotation encoding="application/x-tex">{\displaystyle \Omega _{d}=(d\theta /dt)_{\rm {drift}}}</annotation>

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</math><img src="./d68eff7bb3a2cb8e864d6b5e9363419af1888d91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.702ex; height:2.843ex;" alt="{\displaystyle \Omega _{d}=(d\theta /dt)_{\rm {drift}}}" loading="lazy"> in response to an external torque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{\theta }}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{\theta }}</annotation>

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</math><img src="./55a328bcb0366e512357b01395fc59094aaaaf51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.456ex; height:2.509ex;" alt="{\displaystyle \Gamma _{\theta }}" loading="lazy"> (assuming that the flow stays non-<a href="Turbulence" title="Turbulence">turbulent</a> and that inertial effects can be neglected) is given by

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<annotation encoding="application/x-tex">{\displaystyle \Omega _{d}={\frac {\Gamma _{\theta }}{f_{r}}}}</annotation>

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</math><img src="./16f3f2124f6dcd2c65d0de8176513c41de7b0ac8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.161ex; height:5.676ex;" alt="{\displaystyle \Omega _{d}={\frac {\Gamma _{\theta }}{f_{r}}}}" loading="lazy">,</dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{r}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{r}}</annotation>

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</math><img src="./ed5d68a307fec81cabf96a4c1cfcc0181d5d104c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.113ex; height:2.509ex;" alt="{\displaystyle f_{r}}" loading="lazy"> is the frictional drag coefficient. The relationship between the rotational diffusion coefficient and the rotational frictional drag coefficient is given by the <a href="Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation</a> (or Einstein–Smoluchowski relation):

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</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{r}={\frac {k_{\rm {B}}T}{f_{r}}}}</annotation>

</semantics>

</math><img src="./1fbc912e1956752f1015b1ae09d407958ce63059.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.076ex; height:5.843ex;" alt="{\displaystyle D_{r}={\frac {k_{\rm {B}}T}{f_{r}}}}" loading="lazy">,</dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\rm {B}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{\rm {B}}}</annotation>

</semantics>

</math><img src="./4bd537f082cb894d2f69b1cc050dd36f7d489a3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\rm {B}}}" loading="lazy"> is the <a href="Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> is the absolute temperature. These relationships are in complete analogy to translational diffusion.

The rotational frictional drag coefficient for a sphere of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is

<msub>

<mtext>sphere</mtext>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{r,{\textrm {sphere}}}=8\pi \eta R^{3}}</annotation>

</semantics>

</math><img src="./d314b1299b7e9dd854535d3431a6076f90dc1ae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.731ex; height:3.343ex;" alt="{\displaystyle f_{r,{\textrm {sphere}}}=8\pi \eta R^{3}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

</semantics>

</math><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> is the <a href="Viscosity#Dynamic_(shear)_viscosity" title="Viscosity">dynamic (or shear) viscosity</a>.<a href="#cite_note-9">[8]</a>

The rotational diffusion of spheres, such as nanoparticles, may deviate from what is expected when in complex environments, such as in polymer solutions or gels. This deviation can be explained by the formation of a depletion layer around the nanoparticle.<a href="#cite_note-10">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Langevin_dynamics">Langevin dynamics</h3></div>

Collisions with the surrounding fluid molecules will create a fluctuating torque on the sphere due to the varied speeds, numbers, and directions of impact. When trying to rotate a sphere via an externally applied torque, there will be a systematic drag resistance to rotation. With these two facts combined, it is possible to write the <a href="Langevin_equation" title="Langevin equation">Langevin</a>-like equation:

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dL}{dt}}={I}\,\cdot {\frac {d^{2}{\theta }}{dt^{2}}}=-{\zeta }^{r}\cdot {\frac {d{\theta }}{dt}}+TB(t)}</annotation>

</semantics>

</math><img src="./e5a4c2385ef965c657b248313163576f4ca2917c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:34.88ex; height:6.009ex;" alt="{\displaystyle {\frac {dL}{dt}}={I}\,\cdot {\frac {d^{2}{\theta }}{dt^{2}}}=-{\zeta }^{r}\cdot {\frac {d{\theta }}{dt}}+TB(t)}" loading="lazy">

Where:

<ul><li>L is the angular momentum.</li>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dL}{dt}}}</annotation>

</semantics>

</math><img src="./5ca3bb228fd4f595fca23887a1195167f8485889.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.635ex; height:5.509ex;" alt="{\displaystyle {\frac {dL}{dt}}}" loading="lazy"> is <a href="Torque" title="Torque">torque</a>.</li>

<li>I is the moment of inertia about the rotation axis.</li>

<li>t0 is the start time.</li>

<li>θ is the angle between the orientation at t0 and any time after, t.</li>

<li>ζr is the rotational friction coefficient.</li>

<li>TB(t) is the fluctuating Brownian torque at time t.</li></ul>

The overall Torque on the particle will be the difference between:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle TB(t)}</annotation>

</semantics>

</math><img src="./c7deb7f74710ab9ed9aba32dd34e1bc09ac9903a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.049ex; height:2.843ex;" alt="{\displaystyle TB(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\zeta }^{r}\cdot {\frac {d{\theta }}{dt}})}">

<msup>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\zeta }^{r}\cdot {\frac {d{\theta }}{dt}})}</annotation>

</semantics>

</math><img src="./b268b2349b20f3cad70958b63c91dee16cebb749.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.723ex; height:5.509ex;" alt="{\displaystyle ({\zeta }^{r}\cdot {\frac {d{\theta }}{dt}})}" loading="lazy">.

This equation is the rotational version of <a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newtons second equation of motion</a>. For example, in standard translational terms, a <a href="Rocket" title="Rocket">rocket</a> will experience a boosting force from the engine while simultaneously experiencing a <a href="Drag_(physics)" title="Drag (physics)">resistive force</a> from the air it is travelling through. The same can be said for an object which is rotating.

Due to the random nature of rotation of the particle, the average Brownian torque is equal in both directions of rotation. symbolised as:

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\langle TB(t)\right\rangle =0}</annotation>

</semantics>

</math><img src="./7179c96532e49560af83c9240191765143addc55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.119ex; height:2.843ex;" alt="{\displaystyle \left\langle TB(t)\right\rangle =0}" loading="lazy">

This means the equation can be averaged to get:

<mfrac>

<mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d\left\langle L\right\rangle }{dt}}=-{\zeta }^{r}\cdot \left\langle {\frac {d{\theta }}{dt}}\right\rangle =-{\frac {\zeta ^{r}}{I}}\left\langle L\right\rangle }</annotation>

</semantics>

</math><img src="./b1bfbe0801da8c0fc75be4a39e767e8049a2862c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.752ex; height:6.343ex;" alt="{\displaystyle {\frac {d\left\langle L\right\rangle }{dt}}=-{\zeta }^{r}\cdot \left\langle {\frac {d{\theta }}{dt}}\right\rangle =-{\frac {\zeta ^{r}}{I}}\left\langle L\right\rangle }" loading="lazy">

Which is to say that the first derivative with respect to time of the average Angular momentum is equal to the negative of the Rotational friction coefficient divided by the moment of inertia, all multiplied by the average of the angular momentum.

As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\left\langle L\right\rangle }{dt}}}">

<mfrac>

<mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d\left\langle L\right\rangle }{dt}}}</annotation>

</semantics>

</math><img src="./e4c7a9fb9c247005e675fb8b4101569688a008c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.831ex; height:5.843ex;" alt="{\displaystyle {\frac {d\left\langle L\right\rangle }{dt}}}" loading="lazy"> is the rate of change of angular momentum over time, and is equal to a negative value of a coefficient multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle L\right\rangle }">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\langle L\right\rangle }</annotation>

</semantics>

</math><img src="./096aa3b87202505deb6fe91258c8de2a7980bbc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.392ex; height:2.843ex;" alt="{\displaystyle \left\langle L\right\rangle }" loading="lazy">, this shows that the angular momentum is decreasing over time, or decaying with a decay time of:

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tau {_{L}}}={\frac {I}{\zeta ^{r}}}}</annotation>

</semantics>

</math><img src="./4c78b0dc04ad136c172e43c2ea843053475e7784.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.58ex; height:5.676ex;" alt="{\displaystyle {\tau {_{L}}}={\frac {I}{\zeta ^{r}}}}" loading="lazy">.

For a sphere of mass m, uniform density ρ and radius a, the moment of inertia is:

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I={\frac {2ma^{2}}{5}}={\frac {8{\pi }{\rho }a^{5}}{15}}}</annotation>

</semantics>

</math><img src="./aff6af1a3bcdd82088ae78817e11d300e5ad8503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.508ex; height:5.676ex;" alt="{\displaystyle I={\frac {2ma^{2}}{5}}={\frac {8{\pi }{\rho }a^{5}}{15}}}" loading="lazy">.

As mentioned above, the rotational drag is given by the <a href="Stokes_flow" title="Stokes flow">Stokes</a> friction for rotation:

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\zeta ^{r}}=8\pi \eta a^{3}}</annotation>

</semantics>

</math><img src="./17c1fc6da085bb87eba516ab6f8dc862c8fb9b44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.138ex; height:3.176ex;" alt="{\displaystyle {\zeta ^{r}}=8\pi \eta a^{3}}" loading="lazy">

Combining all of the equations and formula from above, we get:

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tau {_{L}}}={\frac {\rho a^{2}}{15\eta }}={\frac {3}{10}}\tau _{p}}</annotation>

</semantics>

</math><img src="./f5605979c66a6c435b66dad0cb0a3aa2bedcc9b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.317ex; height:6.176ex;" alt="{\displaystyle {\tau {_{L}}}={\frac {\rho a^{2}}{15\eta }}={\frac {3}{10}}\tau _{p}}" loading="lazy">

where:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{p}}</annotation>

</semantics>

</math><img src="./438fec4fcc6c32607b5b4be9b0a9fa3a5aadb247.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.075ex; height:2.343ex;" alt="{\displaystyle \tau _{p}}" loading="lazy"> is the momentum relaxation time</li>

<li>η is the <a href="Viscosity" title="Viscosity">viscosity</a> of the Liquid the sphere is in.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Example:_Spherical_particle_in_water">Example: Spherical particle in water</h3></div>

Let's say there is a virus which can be modelled as a perfect sphere with the following conditions:

<ul><li><a href="Virus" title="Virus">Radius (a) of 100 nanometres</a>, a = 10−7m.</li>

<li>Density: ρ = 1500 kg m−3</li>

<li>Orientation originally facing in a direction denoted by π.</li>

<li>Suspended in water.</li>

<li>Water has a viscosity of η = 8.9 × 10−4 Pa·s at 25 °C</li>

<li>Assume uniform mass and density throughout the particle</li></ul>

First, the mass of the virus particle can be calculated:

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m={\frac {4\rho \pi a^{3}}{3}}={\frac {4\times 1500\times \pi \times (10^{-7})^{3}}{3}}=6.3\times 10^{-18}\mathrm {kg} }</annotation>

</semantics>

</math><img src="./2d1df2633b6fdb6c22a77aa99615c5eaa9c09c5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.857ex; height:5.843ex;" alt="{\displaystyle m={\frac {4\rho \pi a^{3}}{3}}={\frac {4\times 1500\times \pi \times (10^{-7})^{3}}{3}}=6.3\times 10^{-18}\mathrm {kg} }" loading="lazy">

From this, we now know all the variables to calculate moment of inertia:

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I={\frac {2ma^{2}}{5}}={\frac {2\times (6.3\times 10^{-18})\times (10^{-7})^{2}}{5}}=2.5\times 10^{-32}\mathrm {kg} \cdot \mathrm {m} ^{2}}</annotation>

</semantics>

</math><img src="./ddd35163676ac5c0817eed6e10131cc2bb997093.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.443ex; height:5.843ex;" alt="{\displaystyle I={\frac {2ma^{2}}{5}}={\frac {2\times (6.3\times 10^{-18})\times (10^{-7})^{2}}{5}}=2.5\times 10^{-32}\mathrm {kg} \cdot \mathrm {m} ^{2}}" loading="lazy">

Simultaneous to this, we can also calculate the rotational drag:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \zeta ^{r}=8\pi \eta a^{3}=8\times \pi \times (8.9\times 10^{-4})\times (10^{-7})^{3}=2.237\times 10^{-23}\mathrm {Pa} \cdot \mathrm {s} \cdot \mathrm {m} ^{3}}</annotation>

</semantics>

</math><img src="./75e1dd94378f008251d9f68d0c337f9b98d92597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.778ex; height:3.176ex;" alt="{\displaystyle \zeta ^{r}=8\pi \eta a^{3}=8\times \pi \times (8.9\times 10^{-4})\times (10^{-7})^{3}=2.237\times 10^{-23}\mathrm {Pa} \cdot \mathrm {s} \cdot \mathrm {m} ^{3}}" loading="lazy">

Combining these equations we get:

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{L}={\frac {I}{\zeta ^{r}}}={\frac {2.5\times 10^{-32}\mathrm {kg} \cdot \mathrm {m} ^{2}}{2.2\times 10^{-23}\mathrm {Pa} \cdot \mathrm {s} \cdot \mathrm {m} ^{3}}}=1.1\times 10^{-9}\mathrm {kg} \cdot \mathrm {Pa} ^{-1}\cdot \mathrm {s} ^{-1}\cdot \mathrm {m} ^{-1}}</annotation>

</semantics>

</math><img src="./bd3d3ea44066643d6356e7b2d8ffd34713c22184.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:67.222ex; height:6.343ex;" alt="{\displaystyle \tau _{L}={\frac {I}{\zeta ^{r}}}={\frac {2.5\times 10^{-32}\mathrm {kg} \cdot \mathrm {m} ^{2}}{2.2\times 10^{-23}\mathrm {Pa} \cdot \mathrm {s} \cdot \mathrm {m} ^{3}}}=1.1\times 10^{-9}\mathrm {kg} \cdot \mathrm {Pa} ^{-1}\cdot \mathrm {s} ^{-1}\cdot \mathrm {m} ^{-1}}" loading="lazy">

As the <a href="International_System_of_Units" title="International System of Units">SI units</a> for <a href="Pascal_(unit)" title="Pascal (unit)">Pascal</a> are kg⋅m−1⋅s−2 the units in the answer can be reduced to read:

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{L}=1.1\times 10^{-9}\mathrm {s} }</annotation>

</semantics>

</math><img src="./8c41245bad23236d6727690f2a1c5cec967e219a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.852ex; height:3.009ex;" alt="{\displaystyle \tau _{L}=1.1\times 10^{-9}\mathrm {s} }" loading="lazy">

For this example, the decay time of the virus is in the order of nanoseconds.

<div class="mw-heading mw-heading2"><h2 id="Smoluchowski_description_of_rotation">Smoluchowski description of rotation</h2></div>

To write the Smoluchowski equation for a particle rotating in two dimensions, we introduce a probability density P(θ, t) to find the vector u at an angle θ and time t.

This can be done by writing a continuity equation:

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\partial P(\theta ,t) \over \partial t}=-{\partial j(\theta ,t) \over \partial \theta }}</annotation>

</semantics>

</math><img src="./b5f09acd8e897cb2c78d1fe8f15bbbd2b7a44bd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.465ex; height:5.843ex;" alt="{\displaystyle {\partial P(\theta ,t) \over \partial t}=-{\partial j(\theta ,t) \over \partial \theta }}" loading="lazy">

where the current can be written as:

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j(\theta ,t)=-D^{r}{\partial P(\theta ,t) \over \partial \theta }}</annotation>

</semantics>

</math><img src="./8d226a7e64deab31e75d7f94c43e0f42fc6edc75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.027ex; width:22.236ex; height:5.843ex;" alt="{\displaystyle j(\theta ,t)=-D^{r}{\partial P(\theta ,t) \over \partial \theta }}" loading="lazy">

Which can be combined to give the rotational diffusion equation:

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\partial P(\theta ,t) \over \partial t}=D^{r}{\partial ^{2}P(\theta ,t) \over \partial \theta ^{2}}=D^{r}P(\theta ,t)}</annotation>

</semantics>

</math><img src="./b42e926956237d67a3c0e0f7ec655e3df4243419.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:36.937ex; height:6.176ex;" alt="{\displaystyle {\partial P(\theta ,t) \over \partial t}=D^{r}{\partial ^{2}P(\theta ,t) \over \partial \theta ^{2}}=D^{r}P(\theta ,t)}" loading="lazy">

We can express the current in terms of an angular velocity which is a result of Brownian torque TB through a rotational mobility with the equation:

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<annotation encoding="application/x-tex">{\displaystyle j_{B}(\theta ,t)={\dot {\theta }}_{B}P(\theta ,t)}</annotation>

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</math><img src="./5ede819ba06bed2cc9b3943e0c3d1be4d4d5e5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:19.69ex; height:3.343ex;" alt="{\displaystyle j_{B}(\theta ,t)={\dot {\theta }}_{B}P(\theta ,t)}" loading="lazy">

Where:

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}_{B}=\mu ^{r}T_{B}}</annotation>

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</math><img src="./9766d8664aeb4ef79c0807a62f47a6a734f2cdda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.147ex; height:3.343ex;" alt="{\displaystyle {\dot {\theta }}_{B}=\mu ^{r}T_{B}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle T_{B}=-{\partial V_{B} \over \partial \theta }}</annotation>

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</math><img src="./071368be869cfb04c2a010d7027ab4e3c0d37136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.733ex; height:5.509ex;" alt="{\displaystyle T_{B}=-{\partial V_{B} \over \partial \theta }}" loading="lazy"></li>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle V_{B}(\theta ,t)=k_{B}T\ln P(\theta ,t)}</annotation>

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</math><img src="./26b879b483962fc57eece628359f17f19283d43e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.266ex; height:2.843ex;" alt="{\displaystyle V_{B}(\theta ,t)=k_{B}T\ln P(\theta ,t)}" loading="lazy"></li></ul>

The only difference between rotational and translational diffusion in this case is that in the rotational diffusion, we have periodicity in the angle θ. As the particle is modelled as a sphere rotating in two dimensions, the space the particle can take is compact and finite, as the particle can rotate a distance of 2π before returning to its original position

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle P(\theta +2\pi ,t)={P(\theta ,t)}}</annotation>

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</math><img src="./71709995ccfcc01c744fc00a85cdacd18fabb413.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.471ex; height:2.843ex;" alt="{\displaystyle P(\theta +2\pi ,t)={P(\theta ,t)}}" loading="lazy">

We can create a conditional probability density, which is the probability of finding the vector u at the angle θ and time t given that it was at angle θ0 at time t=0 This is written as such:

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P(\theta ,0\mid \theta _{0})=\delta (\theta -\theta _{0})}</annotation>

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</math><img src="./bdcf80a535d0ffc53d4da8e80cf64b8369714e3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.955ex; height:2.843ex;" alt="{\displaystyle P(\theta ,0\mid \theta _{0})=\delta (\theta -\theta _{0})}" loading="lazy">

The solution to this equation can be found through a Fourier series:

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P(\theta ,t\mid \theta _{0})={\frac {1}{2\pi }}\left[1+2\sum _{m=1}^{\infty }e^{-D^{r}m^{2}t}\cos m(\theta -\theta _{0})\right]={\frac {1}{2\pi }}\Theta _{3}({\frac {1}{2}}(\theta -\theta _{0}),e^{-D^{r}t})}</annotation>

</semantics>

</math><img src="./a2a8018016ea5b1fb908bcc89ae4bf5ef524d2b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.343ex; height:7.509ex;" alt="{\displaystyle P(\theta ,t\mid \theta _{0})={\frac {1}{2\pi }}\left[1+2\sum _{m=1}^{\infty }e^{-D^{r}m^{2}t}\cos m(\theta -\theta _{0})\right]={\frac {1}{2\pi }}\Theta _{3}({\frac {1}{2}}(\theta -\theta _{0}),e^{-D^{r}t})}" loading="lazy">

Where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{3}(z,\tau )}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Theta _{3}(z,\tau )}</annotation>

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</math><img src="./aaf8fe1a922d7c0e24768a33b6db25d51b199e46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.996ex; height:2.843ex;" alt="{\displaystyle \Theta _{3}(z,\tau )}" loading="lazy"> is the Jacobian theta function of the third kind.

By using the equation<a href="#cite_note-11">[10]</a>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Theta _{3}(z,\tau )=(-i\tau )^{-1/2}\exp {\biggl (}{\frac {z^{2}}{i\pi \tau }}{\biggl )}\Theta _{3}{\biggl (}{\frac {z}{\tau }},-{\frac {1}{\tau }}{\biggl )}}</annotation>

</semantics>

</math><img src="./06c98d4fb06fc82c32974fb6b1c4cc8a724abdcb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.815ex; height:6.343ex;" alt="{\displaystyle \Theta _{3}(z,\tau )=(-i\tau )^{-1/2}\exp {\biggl (}{\frac {z^{2}}{i\pi \tau }}{\biggl )}\Theta _{3}{\biggl (}{\frac {z}{\tau }},-{\frac {1}{\tau }}{\biggl )}}" loading="lazy">

The conditional probability density function can be written as :

<msub>

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</munderover>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle P(\theta ,t\mid \theta _{0})={\frac {1}{\sqrt {4\pi D^{r}t}}}\sum _{n=-\infty }^{\infty }\exp \left[-{\frac {(\theta -\theta _{0}-2n\pi )^{2}}{4D^{r}t}}\right]}</annotation>

</semantics>

</math><img src="./b11b9427ecff3ddf2b28f15aaa3e18807da5cf7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.24ex; height:7.509ex;" alt="{\displaystyle P(\theta ,t\mid \theta _{0})={\frac {1}{\sqrt {4\pi D^{r}t}}}\sum _{n=-\infty }^{\infty }\exp \left[-{\frac {(\theta -\theta _{0}-2n\pi )^{2}}{4D^{r}t}}\right]}" loading="lazy">

For short times after the starting point where t ≈ t0 and θ ≈ θ0, the formula becomes:

<msub>

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</msub>

<mfrac>

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</mfrac>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\theta ,t\mid \theta _{0})\approx {\frac {1}{\sqrt {4\pi D^{r}t}}}\exp \left[-{\frac {(\theta -\theta _{0})^{2}}{4D^{r}t}}\right]+\cdots }</annotation>

</semantics>

</math><img src="./bf6c24280be3a91e036f9f64d77cd26ca2578554.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.5ex; height:7.509ex;" alt="{\displaystyle P(\theta ,t\mid \theta _{0})\approx {\frac {1}{\sqrt {4\pi D^{r}t}}}\exp \left[-{\frac {(\theta -\theta _{0})^{2}}{4D^{r}t}}\right]+\cdots }" loading="lazy">

The terms included in these are exponentially small and make little enough difference to not be included here. This means that at short times the conditional probability looks similar to translational diffusion, as both show extremely small perturbations near t0. However at long times, t » t0 , the behaviour of rotational diffusion is different to translational diffusion:

<msub>

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</msub>

<mfrac>

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</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle P(\theta ,t\mid \theta _{0})\approx {\frac {1}{2\pi }},t\rightarrow \infty }</annotation>

</semantics>

</math><img src="./0e649a73eba796691f17ca9c383e3880952c028b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.841ex; height:5.176ex;" alt="{\displaystyle P(\theta ,t\mid \theta _{0})\approx {\frac {1}{2\pi }},t\rightarrow \infty }" loading="lazy">

The main difference between rotational diffusion and translational diffusion is that rotational diffusion has a periodicity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta +(2\pi )=\theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta +(2\pi )=\theta }</annotation>

</semantics>

</math><img src="./678ed69129db7137841a8ce080f759a1b6eb7cec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.424ex; height:2.843ex;" alt="{\displaystyle \theta +(2\pi )=\theta }" loading="lazy">, meaning that these two angles are identical. This is because a circle can rotate entirely once before being at the same angle as it was in the beginning, meaning that all the possible orientations can be mapped within the space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">. This is opposed to translational diffusion, which has no such periodicity.

The conditional probability of having the angle be θ is approximately <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2\pi }}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi }}}</annotation>

</semantics>

</math><img src="./129204d50704b07e6a4223870954242b21170354.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.331ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2\pi }}}" loading="lazy"> .

This is because over long periods of time, the particle has had time rotate throughout the entire range of angles possible and as such, the angle θ could be any amount between θ0 and θ0 + 2 π. The probability is near-evenly distributed through each angle as at large enough times.

This can be proven through summing the probability of all possible angles. As there are 2π possible angles, each with the probability of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2\pi }}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi }}}</annotation>

</semantics>

</math><img src="./129204d50704b07e6a4223870954242b21170354.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.331ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2\pi }}}" loading="lazy"> , the total probability sums to 1, which means there is a certainty of finding the angle at some point on the circle.

<ul><li><a href="Diffusion_equation" title="Diffusion equation">Diffusion equation</a></li>

<li><a href="Perrin_friction_factors" title="Perrin friction factors">Perrin friction factors</a></li>

<li><a href="Rotational_correlation_time" title="Rotational correlation time">Rotational correlation time</a></li>

<li><a href="False_diffusion" title="False diffusion">False diffusion</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist">

<li id="cite_note-7"><a href="#cite_ref-7">^</a> This version of the angular diffusion equation assumes spherical symmetry. In a non-symmetric situation, a tensor analog applies.<a href="#cite_note-ZhangDiffusionTensor-6">[6]</a>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">^</a> Conggang Li, Yaqiang Wang, and <a href="Gary_J._Pielak" title="Gary J. Pielak">Gary J. Pielak</a>. The Journal of Physical Chemistry B 2009 113 (40), 13390-13392 DOI:10.1021/jp907744m

</li>

<li id="cite_note-DM-2"><a href="#cite_ref-DM_2-0">^</a> <a href="David_Merritt" title="David Merritt">Merritt, D.</a> (2002), <a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/2002ApJ...568..998M">Rotational Brownian Motion of a Massive Binary</a>, The Astrophysical Journal, 568, 998-1003. Retrieved 28 March 2022

</li>

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</style><cite id="CITEREFPerrin1934" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Francis_Perrin_(physicist)" title="Francis Perrin (physicist)">Perrin, Francis</a> (1934). <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/jpa-00233265/document">"Mouvement brownien d'un ellipsoide (I). Dispersion diélectrique pour des molécules ellipsoidales"</a>. <a href="Journal_de_Physique" class="mw-redirect" title="Journal de Physique">Journal de Physique</a> (in French). 7 (5): 497–511. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1051%2Fjphysrad%3A01934005010049700">10.1051/jphysrad:01934005010049700</a>.</cite>

</li>

<li id="cite_note-perrin_1936-4"><a href="#cite_ref-perrin_1936_4-0">^</a> <cite id="CITEREFPerrin1936" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Francis_Perrin_(physicist)" title="Francis Perrin (physicist)">Perrin, Francis</a> (1936). <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/jpa-00233379/document">"Mouvement brownien d'un ellipsoide (II). Rotation libre et dépolarisation des fluorescences: Translation et diffusion de molécules ellipsoidales"</a>. Le Journal de Physique (in French). 7 (7): 1–11. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1051%2Fjphysrad%3A01936007010100">10.1051/jphysrad:01936007010100</a>.</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFMichalskiKalwarczykKwapiszewskaEnderlein2024" class="citation journal cs1">Michalski, Jarosław; Kalwarczyk, Tomasz; Kwapiszewska, Karina; Enderlein, Jörg; Poniewierski, Andrzej; Karpińska, Aneta; Kucharska, Karolina; Hołyst, Robert (2024-07-11). <a rel="nofollow" class="external text" href="https://doi.org/10.1039%2FD4SM00422A">"Rotational and translational diffusion of biomolecules in complex liquids and HeLa cells"</a>. Soft Matter. 20 (29): 5810–5821. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2024SMat...20.5810M">2024SMat...20.5810M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1039%2FD4SM00422A">10.1039/D4SM00422A</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1744-6848">1744-6848</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/38995242">38995242</a>.</cite>

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<li id="cite_note-ZhangDiffusionTensor-6"><a href="#cite_ref-ZhangDiffusionTensor_6-0">^</a> <cite id="CITEREFZhangZhaoCao2019" class="citation journal cs1">Zhang, Zi-Tong; Zhao, Xin; Cao, Bing-Yang (12 December 2019). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6908649">"Diffusion Tensors of Arbitrary-Shaped Nanoparticles in Fluid by Molecular Dynamics Simulation"</a>. Scientific Reports. 9 (1): 18943. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019NatSR...918943Z">2019NatSR...918943Z</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41598-019-55042-9">10.1038/s41598-019-55042-9</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6908649">6908649</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/31831762">31831762</a>.</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFJones" class="citation web cs1">Jones, Robert. B. <a rel="nofollow" class="external text" href="https://rcin.org.pl/Content/67140/WA727_24860_56182_Jones-Rotational.pdf">"Rotational Diffusion in Dispersive Media"</a> (PDF). Warsaw, Poland: Institute of Fundamental Technological research. p. 21. Retrieved 16 March 2022.</cite>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFMaldonado-CamargoYangRinaldi2017" class="citation journal cs1">Maldonado-Camargo, Lorena; Yang, Chuncheng; Rinaldi, Carlos (2017-08-24). "Scale-dependent rotational diffusion of nanoparticles in polymer solutions". Nanoscale. 9 (33): 12039–12050. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1039%2Fc7nr01603d">10.1039/c7nr01603d</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2040-3372">2040-3372</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28795729">28795729</a>.</cite>

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFCantorSchimmel_PR1980" class="citation book cs1">Cantor, CR; Schimmel PR (1980). Biophysical Chemistry. Part II. Techniques for the study of biological structure and function. W. H. Freeman.</cite></li>

<li><cite id="CITEREFBerg1993" class="citation book cs1">Berg, Howard C. (1993). Random Walks in Biology. Princeton University Press.</cite></li></ul>

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