Active matter refers to a type of bulk matter, often soft condensed matter that is an Active system, i.e. that it produces its own driving energy (for example, self-propelling particles, and micro-swimmers).

Driven matter is a closely related type of matter, where the system is externally driven.

The Hydrodynamics of Active Systems

Life at low Reynold's numbers, see Low Reynolds number

topics on active and soft matter physics assignment – Self-assembly of active colloids

See also Complex fluid dynamics, Colloid physics

Single swimmer hydrodynamics: background

Swimming at low Reynolds number: Stokes equation

Important consequence for swimmers: Scallop theorem (see Kinematic reversibility in fluid dynamics)

Swimmer models

• Squirmer. _{Swimming efficiency of spherical squirmers: Beyond the Lighthill theory}

Far-flow fields

_{Used to calculate hydrodynamic interactions, for instance}

Point-force problem for Stokes equation, can be solved using its Green function, often called the Oseen tensor (see here):

$G_{ij}(\vec{r}) = \frac{1}{8\pi \mu} \left(\frac{\delta_{ij}}{|\vec{r}|}+\frac{r_i r_j}{|\vec{r}|^3}\right)$

_{where the $\frac{1}{8\pi \mu}$ is often omitted in the definition of the Oseen tensor.}

Using the Green function to construct general solution, one can construct a multipole expansion. As swimmers (in average, in steady state) don't accelerate, the fluid isn't exerting a net force on them, so they can't be exerting a net force on the fluid (Netwon's third law). Therefore the monopole term (called the Stokelet) isn't present. _{An exception for this is the relatively large microorganism Volvox, for which gravity force is significant, giving a net force to the problem and creating a Stokelet flow}. Therefore, the dominant term is generally the dipolar term:

$v_i (\vec{r}) \approx \frac{\partial G_{ij}}{\partial x_k} (\vec{r}) D_{jk}$

where $v_i (\vec{r})$ is the velocity field, and

$D_{jk} = - \int f_j \xi_k d\vec{\xi}$

It is conventional to let:

$D_{jk} \rightarrow D_{jk} -\frac{1}{3}D_{ii} \delta_{jk} \equiv S_{jk} + T_{jk}$

where the addition of $\frac{1}{3}D_{ii}$ doesn't change the velocity field $v_i (\vec{r})$ because $\vec{\nabla} \cdot \mathbf{G} = \vec{0}$. $S_{jk}$ is called the stresslet, and $T_{jk}$ is called the rotlet. The rotlet is zero if the net torque on the fluid is zero, which it is for active microswimmers.

Note that the dipolar flow has nematic symmetry; this is important in the collective behavior of active swimmers.

We can have two kinds of dipolar flow around a swimmer:

• pusher, or extensive . Like that of the E. Coli
• puller, or contractile Like that of the Chlamydomonas. See figure of chlamydomonas flow

chlamydomonas flow source

Single microswimmer hydrodynamics: applications

• bacteria enhance diffusion as a result of the flow fields they produce
• motion of swimmers in background/external flow.
• interactions with surfaces.

Collective hydrodynamics of active entities

• Beris-Edwards equations
• extra stress from active particles, equals the average value of the stresslet, and gives the active stress $-\zeta Q_{jk}$
• Different kinds of instabilities and patterns arise.

Collective hydrodynamics of active entities: applications

• active turbulence
• interactions between topological defects, walls (regions of high bend perturbation), and flows (jets, and vortical).
• Lyotropic active nematics and active anchoring
• Example system: microtubules and Molecular motors.

Other applications

• More general Active systems and types of active matter: dry systems, systems with polar symmetry, density variations, inertia.
• Active machines and Self-assembly
• Microswimmers moving in a viscoelastic medium. Living liquid crystals represent a novel system where bacteria swim in a nematic liquid.
• Biological systems (see Biological matter):
  • molecular motors walking along microtubules contribute to cell division resulting from spindle mitosis.
  • cytoplasmic streaming, flow driven by the motion of motors along the cell walls, presumably to aid the transport of nutrients around the cell.
  • The extent to which hydrodynamics (even at nanometre scales) affects motor motion [74], the way in which multiple motors can combine to move cargo and mechanisms for cargo transport in the crowded cellular environment remain largely unexplored.
  • there is increasing evidence that cell motility is linked to the physical environment. Interactions between cells, the spreading of cellular layers and the possible role of flow in Morphogenesis are also of interest.
• Active gel physics

Physics of Microswimmers – Single Particle Motion and Collective Behavior

In pursuit of propulsion at the nanoscale

Biphasic, Lyotropic, Active Nematics

Papers on active matter

Self organization in active matter

Cytoplasmic streaming

A physical perspective on cytoplasmic streaming

Cytoplasmic streaming in plant cells emerges naturally by microfilament self-organization

Spontaneous Circulation of Confined Active Suspensions

Instabilities, pattern formation, and mixing in active suspensions

Spindle self-organization

Physical basis of spindle self-organization