### What is anomalous transport?

This is a new, exciting area of research because it is a widespread natural phenomenon.
Examples include flight of albatross, human migration, movement of proteins on cell membranes and signalling molecules in the brain,
transport on social networks and fractal geometries,
bacterial motility. Anomalous transport cannot be described by standard tools like diffusion equation. Instead it requires the use of
fractional partial differential equations involving fractional derivatives of non-integer order.
Diffusion processes in complex systems may fail to obey linear law for unforced diffusion \( \langle x^2(t) \rangle \sim t \).
They may instead obey a power law pattern of growth of the mean squared displacement \( \langle x^2(t) \rangle \sim t^\gamma \).

Anomalous subdiffusive systems observe ultra-slow transport with an evolution of mean squared displacement as \( \langle x^2(t) \rangle \sim t^\gamma \) for \( 0 < \gamma < 1 \).
The relaxation is governed by the Mittag-Leffler function, which interpolates between an initial stretched exponential and asymptotic power law.
The Mittag-Leffler function is an exact solution of the fractional relaxation equation \( \frac{d \Phi(t)}{dt} = -\tau_0 D_t^{1-\gamma} \Phi(t) \), where \( D_t^{1-\gamma} \) is the Riemann-Liouville fractional derivative operator.

Anomalous diffusion is an observed natural phenomenon, found in many varied areas of science including: transport of lipids on cell membranes, bacterial motion, transport on fractal geometries, NMR diffusometry in porous systems, tracer diffusion in subdiffusive hydrology, signalling molecules in spiny dendrites, RNA molecules in cells, flight of albatross, financial futures prices, quantum optics, dispersive transport in amorphous semiconductors, and others.

Subdiffusion is classically described in terms of the continuous time random walk (CTRW) with an asymptotic power law waiting time of the form \( \psi(t) \sim \tau_0 / t^{1 + \gamma} \) for \( 0 < \gamma < 1 \) and \( t \gg \tau \).
This is characterised by a diverging first moment, ie. an infinite mean waiting time for walkers.
Such waiting times introduce deeply non-Markovian behaviour into the CTRW, where the current state of the system now depends on the entire history from preparation.

From the CTRW model with anomalously long waiting times one can recover, in the diffusion limit, the fractional Fokker-Planck equation for position, time pdf \( p(x,t) \)
$$ \frac{\partial p}{\partial t} =\mathcal{D}_t^{1-\gamma}K_\gamma\left[ \frac{\partial^2}{\partial x^2}-\frac{\partial }{\partial x}\frac{F(x)}{k_BT}\right]p(x,t)$$

### What do we do?

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We are interested in

statistical mechanics of non-equilibrium processes,
random walk theory,
anomalous subdiffusive and
superdiffusive transport in biophysical systems ,
travelling wave theory and human migration
modelling of intracellular transport.
We are also interested in the study of chemotaxis, anomalous aggregation within a subdiffusive environment,
where the diffusion is influenced by a gradient of concentration of an external chemical signal.
In inhomogeneous media, under the CTRW framework, the escape rate of a random walker from a point is not constant throughout the domain.
This introduces a dependence of the anomalous exponent on the spatial co-ordinate.
Any spatial fluctuation in the anomalous exponent can lead to the phenomenon of anomalous aggregation.
Anomalous aggregation can be negated by a tempering of the anomalous waiting time.
We are interested in studying the interaction between attractive forces such as anomalous aggregation,
and possible repulsive forces such as nonlinear reactions and chemotactic forces.

We are studying the transport of all kinds of components within the cell - from vesicles along the cytoskeleton through to transcription
factors along DNA.
Despite experimental facts that intracellular transport is heterogeneous and non-Markovian with subdiffusive and superdiffusive regimes
most mathematical models for vesicles trafficking are Markovian and homogeneous.
The main challenge for our Manchester interdisciplinary team is to obtain new non-Markovian models of heterogeneous
intracellular transport supported by experiments. These models will provide a tool set for analysing transport processes in a much more
realistic way, opening the way for greatly improved analysis and ultimately understanding of
these highly complex cellular behaviours. This will allow other researchers to formulate and test new hypotheses. In the long term,
therefore, non-Markovian models have the potential to lead to insight into neurological diseases, ageing and other processes
that involve intracellular transport such as bacterial and viral infection. Such knowledge will be important for developing new treatments.
Our project combines three different approaches: mathematical modelling, numerical modelling and experimental validation,
which complement each other. This strategy will provide multidisciplinary study of the intracellular transport problem
and ensure maximum impact across and within several disciplines.