We consider the dynamics of Brownian particles exposed to symmetric periodic potentials and driven out of equilibrium by symmetric driving forces at several examples. For any non-zero temperature, the average velocity vanishes but there may be dynamical states of sustained transport spontaneously breaking the symmetry in the deterministic limit, named spontaneous symmetry breaking transport. If the symmetry of the dynamics is broken by a small perturbation, e.g. a constant bias force, the particle mobility may be in practically any direction, even in the direction opposite to the bias force. Changing only a scalar parameter of the dynamics, e.g. the amplitude or frequency of the symmetric driving force, the particle can be directed into almost any direction. Without a constant bias force, the diffusion coefficient of the particle diverges in the deterministic limit. In more than one (spatial) dimension, diffusion is highly anisotropic, and the direction of fast diffusion can be adjusted by a scalar parameter of the dynamics, e.g. the amplitude of the symmetric driving force. Our examples are (i) the dynamics of an underdamped Brownian particle in a one dimensional periodic potential, published in [1, 2] and verified experimentally in [3], (ii) the dynamics of an overdamped Brownian particle in a two dimensional square lattice, published in [4], and (iii) the dynamics of two interacting Brownian particles, i.e. a dimer, in a one dimensional periodic potential. Lastly, we show how lattice potentials can be exploited to sort molecules that differ only by their chirality,published in [5]. Using spontaneous breaking of chiral symmetry, the two chiral partners can be directed into orthogonal directions under the influence of a constant bias force in a symmetry direction of the lattice and into opposite directions if a periodic driving force is applied.

[1] D. Speer, R. Eichhorn, and P. Reimann, "Brownian motion: Anomalous response due to noisy chaos," Europhys. Lett., vol. 79, p. 10005, 2007.

[2] D. Speer, R. Eichhorn, and P. Reimann, "Transient chaos induces anomalous transport properties of an underdamped Brownian particle," Phys. Rev. E, vol. 76, p. 051110, 2007.

[3] J. Nagel, D. Speer, T. Gaber, A. Sterck, R. Eichhorn, P. Reimann, K. Ilin,M. Siegel, D. Koelle, and R. Kleiner, "Observation of negative absolute resistance in a Josephson junction, "Phys. Rev. Lett., vol. 100, p. 217001, 2008.

[4] D. Speer, R. Eichhorn, and P. Reimann, "Directing Brownian motion on a periodic surface," Phys. Rev. Lett., vol. 102, p. 124101, 2009.

[5] D. Speer, R. Eichhorn, and P. Reimann, "Exploiting lattice potentials for sorting chiral particles," Phys. Rev. Lett., vol. 105, p. 090602, 2010.