Dynamics of an active magnetic particle in a rotating magnetic ﬁeld A. Cēbers * Institute of Physics, University of Latvia, Salaspils-1, LV-2169, Latvia M. Ozols University of Latvia, Zellu-8, Riga, Latvia Received 5 September 2005; revised manuscript received 21 October 2005; published 8 February 2006 The motion of an active self-propelling particle with a permanent magnetic moment under the action of a rotating magnetic ﬁeld is considered. We show that below a critical frequency of the external ﬁeld the trajec- tory of a particle is a circle. For frequencies slightly above the critical point the particle moves on an approxi- mately circular trajectory and from time to time jumps to another region of space. Symmetry of the particle trajectory depends on the commensurability of the ﬁeld period and the period of the orientational motion of the particle. We also show how our results can be used to study the properties of naturally occurring active magnetic particles, so-called magnetotactic bacteria. DOI: 10.1103/PhysRevE.73.021505 PACS numbers: 83.80.Gv, 47.15.G-, 87.19.St Active systems interacting with electromagnetic ﬁelds are interesting from different points of view 1,2. The pumping of liquid without force due to the negative viscosity effect of dielectric suspensions 3, the ﬂexible magnetic ﬁlaments in- teracting with an ac magnetic ﬁeld 4,5, and their self- propulsion in liquid 6,7 may also be mentioned here among other examples. Active magnetic systems also exist in nature—the magnetotactic bacteria have a biochemical ma- chinery allowing them to produce ferromagnetic particles in- side their bodies and use the particles to orientate in the magnetic ﬁeld of the Earth 8,9. Active magnetic systems have very interesting properties which have not been completely investigated yet. In this pa- per we consider an active particle with a permanent magnetic dipole under the action of a rotating magnetic ﬁeld. The study of the behavior of magnetic particles in a rotating mag- netic ﬁeld has a rather long history see, for example, Ref. 10 for further references. Among the most recent develop- ments in this ﬁeld we can mention Ref. 11, where the mo- tion of an anisotropic particle in the rotating optical ﬁeld is investigated. Several regimes of the particle motion in a ro- tating magnetic ﬁeld are established—synchronous motion at frequencies below the critical and back and forth motion at frequencies above it 12. Interesting features occur in the behavior of ﬂexible magnetic particles under the action of the rotating ﬁeld 13,14. In spite of this long-standing inter- est in the behavior of different particles in ac magnetic ﬁelds, active particles self-propelled in a liquid have never been properly investigated. An indication that a lot of interesting things can take place in this case can be found in the paper 15, where from the ﬁgures one can see that at the frequen- cies above the critical frequency the character of magneto- tactic bacteria motion changes drastically. Instead of motion along the circles which occurs if the frequency is low enough, the bacteria starts to jump between the circles of a smaller radius at higher frequencies. Such a regime of the particle motion has never been studied before. Here we present a simple model of the behavior of an active particle with a permanent magnetic moment in a rotating magnetic ﬁeld. It turns out that a simple combination of active prop- erties of a particle with its capability to orientate along the applied ﬁeld leads to rather rich behavior which has interest- ing possibilities for different practical applications among which the determination of the physical properties of the magnetotactic bacteria should be mentioned. Let us introduce our model. The magnetic particle, due to its self-propulsion, moves with velocity v in the direction of its magnetic moment this mechanism is used by magneto- tactic bacteria for their survival in the environment 8,9,16. Keeping in mind magnetotactic bacteria, as the example, we do not consider the possibility of their tumbling 17. Al- though some of them have only one ﬂagella, nevertheless there are exceptions 18. Inﬂuence of tumbling on the mo- tion of magnetotactic bacteria remains an interesting issue to study in the future. Besides this the active particle is approxi- mated by an axisymmetric body. It should be noted that heli- coidal trajectories are observed for some magnetotactic bac- teria due to their nonaxisymmetry 18. Its inﬂuence on motion of bacteria in the rotating ﬁeld is an interesting issue to study in the future also. The coupling of the translational and rotational degrees of freedom of the bacteria are essen- tial, for example, in their motion near solid walls 19. The magnetic moment orientates along the applied ﬁeld H. The kinetics of orientation is determined by the torque balance acting on the particle, -  d dt + MH sin  =0. 1 Here M is the dipole moment of the particle,  is the rota- tional friction coefﬁcient,  is the particle orientation angle with respect to some ﬁxed direction, which we take to be the x axis, and  is the angle between the magnetic ﬁeld and magnetic moment of the particle. Inﬂuence of the thermal ﬂuctuations on the orientational dynamics of the particle is *Electronic address: aceb@tesla.sal.lv PHYSICAL REVIEW E 73, 021505 2006 1539-3755/2006/732/0215055/$23.00 ©2006 The American Physical Society 021505-1