Non-Equilibrium, Weak-Field Induced Magnetism: a Mechanism For

Non-equilibrium, weak-ﬁeld induced magnetism: a mechanism for magnetobiology

Ashot Matevosyan1) and Armen E. Allahverdyan2)

1)University of Cambridge, Cavendish Laboratory, 19 J.J. Thompson avenue, Cambridge CB3 0HE, UK, 2)Alikhanyan National Laboratory (Yerevan Physics Institute), Alikhanian Brothers Street 2, Yerevan 375036, Armenia

February 2, 2021

Abstract There is a long-time quest for understanding physical mechanisms of weak magnetic field interaction with biological matter. Two factors impeded the development of such mechanisms: first, a high (room) temperature of a cellular environment, where a weak, static magnetic field induces a tiny (classically zero) equilibrium response. Second, the friction in the cellular environment is very large, preventing a weak field to alter non-equilibrium processes such as a free diffusion of charges. Here we study a class of non-equilibrium steady states of a cellular ion in a confining potential, where the response to a (weak, homogeneous, static) magnetic field survives strong friction and thermal fluctuations. The magnetic field induces a rotational motion of the ion that proceeds with the cyclotron frequency. In particular, such non-equilibrium states are generated by a white noise acting on the ion additionally to the (non-local) friction and noise generated by an equilibrium thermal bath. Keywords: static magnetic field; Langevin equation; non-equilibrium steady states

1 Introduction

The influence of a weak, static magnetic field 1 on biological systems remains a controversial subject [1,2,3]. A weak diamagnetism is present in such systems [2], but it can produce visible effects only for high magnetic fields (∼ 20 T)[4]. Experimental reports on the existence of weak-field biological influences are too numerous to be wrong [1,3]. But they are frequently not reproducible, which by itself asks for an explanation [1,3]. On the other hand, physical mechanisms that would describe such influences in a sufficiently general molecular biology situation are unclear [5,6,7,8,9]. One of few cases, where the influence mechanism was clarified includes magnetotaxis phenomena, where certain bacteria are able to sense the Earth magnetic field via synthesizing ferromagnetic particles [2]. They respond to weak magnetic fields for the same reason as the compass responds to the Earth magnetic field, viz. due to a strong ferromagnetic interaction between the magnetic moments. Another situation, where a well-known physical effect is applied to magnetobiology includes (bio)chemical reactions with radical pairs [10, 11, 12, 13, 14]. Here the reaction rate depends on spins of reacting radicals, whose coherent quantum dynamics involves quasi-degenerate energy levels and hence may be susceptible to weak magnetic fields. The necessary condition for this effect arXiv:2102.00481v1 [cond-mat.stat-mech] 31 Jan 2021 is that the environmental influences (dephasing and relaxation) acting on spins are negligible for reaction times. The radical pair mechanisms are investigated in the context of several scenarios of animal magnetoreception [2], also via modern tools of quantum physics [15, 16, 17]. However,

1 The conventional division of static magnetic field magnitudes is as follows [1]: weak (< 1 mT), moderate (> 1 mT and < 1 T), high (> 1 T and < 20 T), ultra-high (> 20 T). Modern NMR medicine operates in the strong field range 1 − 3 T, while the Earth magnetic field is ∼ 50µT.

1 these mechanisms cannot apply for describing all magnetobiological phenomena [7]; they were so far established in vitro only [12], and they also suffer from fragility and non-reproducibility [13, 14]. Here we plan to study the influence of a weak static magnetic field on the stochastic motion of an ion (Brownian charge). Metal ions (Na+, K+, Ca2+, Mg2+ etc) play an important role in molecular biology. They can be called its third ingredient along with DNA/RNA and proteins [18]. Nearly 1/3 of all proteins employ metal ions for their functioning [2, 18]. Ions are important in bioenergetics, communication (e.g. nerve impulse generation), osmotic regulation etc [2, 18, 19]. Hence it is natural to study their motion as a target of a static magnetic field. Such a study should naturally look at non-equilibrium states, since under weak magnetic fields ions translational motion is classical, and then an equilibrium magnetic response (moment) is excluded by the Bohr- van Leeuwen theorem [20, 21, 22, 23, 24, 25]. However, the main issue of envisaging possible non-equilibrium responses of a static magnetic field is that there is a huge difference between the friction-induced inverse relaxation time of ions in water and the magnetic (cyclotron) frequency of weak fields [5,6,7,8]. For example, consider the free ion diffusion (i.e. Brownian motion without an external potential) that is a pertinent non-stationary (hence non-equilibrium) cellular process [2]. Theoretically, it is influenced by a static magnetic field, since the diffusion is impeded in directions perpendicular to the field due to a circular (cyclotron) motion induced by the magnetic field [26, 27]. But a for weak fields this effect is completely diminished by large translation friction (damping) in water: the influence of the magnetic field on the perpendicular diffusion is or order of b2/γ2, where b and γ are (resp.) the cyclotron frequency induced by the magnetic field and the friction coefficient in water. For the Earth magnetic field we get b2/γ2 ∼ 10−20; see section 2.4 for similar estimates. Models that attempt to explain the effects of weak magnetic fields postulate the existence of weakly-damped degrees of freedom [3, 28, 29, 30, 31, 32, 33]. However, physical estimates show that possible differences in the value of the friction (e.g. those due to hydrophobic effects) are far too small to account for the above huge difference between the friction and the cyclotron frequency [5,6,7,8,9]. We aim to propose mechanisms for the influence of weak static magnetic fields that survives strong friction and room temperature fluctuations, i.e. they lead to ion rotation with the cyclotron frequency in a weak field and an aqueous environment. These mechanisms involve violations of Fluctuation-Dissipation Relation (FDR) that produce non-equilibrium stationary states, where the Bohr-van Leeuwen theorem does not generally hold [23, 24, 25]. The existence of FDR violations is well-established in active biophysical processes that involve external energy input (e.g. from ATP); see [34, 35] for reviews. While it is frequently difficult to distinguish situations without FDR from those where it holds (both can be noisy and frictional) [34], the difference between them is a conceptual one, since violations of FDR is the hallmark of non-equilibrium. Here we shall focus on one specific scenario of FDR violation that relates to cellular noise [36, 37]. The relation of the magnetic response to non-equilibrium physics allows to hypothesize about the incomplete reproducibility of magnetobiological experiments. It can be related to the fact that non-equilibrium states are not self-sustainable and may change during the experimental time. The rest of this paper is organized as follows. Section2 discusses our basic model: an ion interacting with an equilibrium thermal bath and subjected to an external white noise. Section3 solves this model in (an)harmonic confining potential and demonstrates its diamagnetic response to an external magnetic field. In section 3.3.1 we study the influence of thermal fluctuations on this diamagnetism and show that it relates to a signal-to-noise ratio which is comparable to e.g. with that of electric current in a good conductor. Section 3.3.2 shows that though the (classical) equilibrium state does not posses an average magnetic moment (Bohr-van Leeuwen theorem), certain traces of magnetic field survive even in such a state and can be amenable to indirect observation. Section4 underlines a related model that can show a paramagnetic response. Section5 visualizes stochastic trajectories for several ranges of parameters and explains them via autocorrelation functions. We summarize in the last section. Several issues are relegated to Appendices. In particular, AppendixA compares the obtained non-equilibrium classical magnetic moments with known scenarios of equilibrium and dissipative quantum magnetism, while Appendix B describes our main technical method.

2 2 The model 2.1 Langevin equation We shall mostly study ions bound in external potentials (generated by membranes or proteins). Hence we shall neglect electrostatic interaction between ions [2], focusing instead on their interaction with water thermal bath and with external forces. Thus we describe a single ion with mass dx m, charge Q, coordinate vector x(t) = (x, y, z), and velocity v = x˙ ≡ dt . The Langevin equation divided over the mass m reads [38, 39]:

Z t 0 −κ|t−t0| 0 Q 1 1 v˙ (t) = −γ dt κ e v(t ) − ux(x) + v × B + η(t) + ξ(t), (2.1) t0 m m m where Q v × B is Lorentz’s force coming from an external static magnetic ﬁeld B (here × stands for the vector product), ux(x) ≡ ∂u/∂x is the force generated by a potential u(x), and where the friction force with magnitude γ and exponential kernel has a well-deﬁned memory time 1/κ. In (2.1) η(t) and ξ(t) are independent Gaussian noises. The initial conditions for (2.1) are posed at t = t0. The parameters γ, κ and Q|B|/m in (2.1) have dimension of frequency. Here Q|B|/m is the well-known cyclotron frequency [1,3], and the proper friction constant is mγ. At least one additional frequency parameter comes from u(x), e.g. the isotropic harmonic potential brings frequency ω0: ω2 ω2 u(x) = 0 |x|2 = 0 x2 + y2 + z2 . (2.2) 2 2

Let δ(t) and δij be Dirac’s delta-function and Kronecker’s symbol, respectively. The averages h...i of the Gaussian noises read:

hηi(t)i = 0, hξi(t)i = 0, i = x, y, z, (2.3) 0 θ −θ|t−t0| 0 0 hηi(t)ηj(t )i = δij q 2 e , hξi(t)ξj(t )i = qwδijδ(t − t ). (2.4)

Here q and qw are noise intensities, 1/θ is the correlation time of the η-noise, while the ξ-noise is white. Recall that if in (2.1) the friction (∝ γ) and the noise η(t) are generated by the same equilibrium thermal bath, then the ﬂuctuation-dissipation relation (FDR) holds [38], i.e. the friction 0 0 memory kernel e−κ|t−t | and the noise correlation kernel e−θ|t−t | are the same:

θ = κ. (2.5)

Another aspect of the FDR is that the intensity of the η(t)-noise relates to the thermal bath temperature T (kB is Boltzmann’s constant):

q = 2mγkBT. (2.6)

We shall frequently (but not always) assume that (2.5, 2.6) hold, and then the only non-equilibrium aspect in the problem comes from the external white noise ξ(t) that does not satisfy FDR, since it is not accompanied by the corresponding friction. Equations similar to (2.1), i.e. Langevin equations with magnetic ﬁeld, were studies at many places in application to plasma physics or stochastic systems [22, 23, 24, 27, 40, 41, 42, 43]; see especially [25, 44, 45, 46]. Our focus will be on implications of (2.1) for magnetobiology, i.e. for ions moving in cellular environment having a large friction (γ ∼ 1012 s−1) and strong thermal ﬂuctuations (T = 300 K).

2.2 Memory and noise The proper understanding of memory times in friction (and noise) is crucial for our approach. The 0 friction in (2.1) has a ﬁnite memory time 1/κ. For κ → ∞ we get κe−κ|t−t | → 2δ(t − t0), and then the friction in (2.1) reverts to its usual, memoryless form −γv [38, 39]. For the Brownian ion

3 moving in a dense fluid (water) the friction (i.e. energy dissipation) is generated by fluid’s shear viscosity [47, 48, 38]. The general physical mechanism for the memory in friction is that relaxation processes in the fluid have a finite characteristic time, which (roughly) agrees with the memory time of the friction acting on the ion [47, 48]. For the ion with mass mi the friction constant miγi can be estimated via the Stokes-Einstein formula [47, 48]:

miγi ∼ riη, (2.7)

−2 where ri is the (effective) radius of the ion, and η is the shear viscosity of the fluid; η = 10 g cm/s for water at room temperature. A similar formula γf mf ∼ rf η can be written for the fluid molecule, which renders its relaxation time mf /(rf η). Now the memory time 1/κ of the ion can be estimated as 1/κ ∼ mf /(rf η) [48]. This memory time is much smaller than the relaxation time mi/(riη) of the ion for mf mi, which is a known condition for the memoryless friction [47, 48]. + + 2+ For biological ions (Na , K , Ca ) moving in the water we have mf ∼ mi (and rf ∼ ri), hence the memory in friction should not be neglected, as predicted theoretically (see [48] for a review) and 1 observed experimentally [49, 50]. While this estimate for κ apply to any fluid, water has specific 1 features that make the real estimate for κ somewhat larger; see section 2.4 for details. Note that the hydrodynamic theory of the friction memory leads to a much more complex expression compared to the exponential form in (2.1). This theory is not yet complete [51] and 0 contains several open questions. Hence we choose the exponential form e−κ|t−t | as a useful heuris- tics that reproduces well the rigidly observed facts of the free Brownian motion, i.e. the ballistic motion at small times and the ordinary diffusion at larger times [38, 49, 50, 48]. The virtue of exponential kernels is that below we can construct from (2.1) a Fokker-Planck equation that applies to anharmonic external potentials. Once the memory in friction is understood, the structure of the η-noise in (2.5) is recovered through the FDR (2.5). The ξ-noise in (2.1) emerges due to various cellular processes: protein production and degradation, the opening and closing of ion channels, molecular diffusion, etc [36, 37]. Such processes are non-equilibrium, and hence ξ does not hold FDR, i.e. it is not accompanied by friction. Cellular noises tend to average out for objects of size & 1 µ m [36], but they are certainly important for smaller objects, e.g. ions, whose size is less than ∼ 1 n m [36]. We adopted a simplifying assumption that ξ is a white Gaussian noise with the magnitude qw. The ξ-noise will not make big differences for certain stationary features of ion’s motion, once qw comparable or smaller than the magnitude q of the thermal noise η. But it will be important for e.g. the orbital magnetic moment of the ion, as we show below. The assumption on the white feature of ξ is not mandatory in our approach and will be relaxed in section4.

2.3 Fokker-Planck equation We are interested in long-time (stationary) averages generated by (2.1). For reaching the stationary state, it is necessary and sufficient that the potential u(x) is confining, i.e. it goes to infinity for |x| → ∞ [38, 39]. Moving from (2.1) to the Fokker-Planck equation is not straightforward, since (2.1) is an integro-differential equation and contains the non-white noise η(t). To facilitate the construction of the Fokker-Planck equation below, we represent the friction in (2.1) via a linear, first-order in time equation that contains an additional variable r. To this end, define

t Z 0 r ≡ γ dt0 κ e−κ|t−t | v(t0), (2.8) t0 r˙s = −κ rs + γκ vs, s = x, y, z. (2.9)

Solving (2.9) with initial conditions at t = t0 and taking t0 → −∞ we get back (2.8). The limit t0 → −∞ is a natural one, since we are eventually interested in the stationary state. Likewise, the noise ηs (s = x, y, z) in (2.4) is represented for t0 → −∞ as a solution of a linear, ﬁrst-order in time diﬀerential equation with white noise εs(t):

√ 0 η˙s = −θηs + θ q εs(t), hεs(t)i = 0, hεs(t)εs0 (t)i = δss0 δ(t − t ). (2.10)

4 QB We deﬁne b = m as the cyclotron frequency. From this point, for simplicity take m = 1 and Q = 1. To recover the solution containing mass (m) and charge (Q), replace in formulas below q q QB q → , q → w , b → . (2.11) m2 w m2 m Together with the new variables introduced in (2.9, 2.10), the Langevin equation (2.1) reads after putting B = bez and using (2.10, 2.8):

v˙ = −r + b v × ez − ux + η + ξ, (2.12) or after projecting it on the (x, y, z) plane,

v˙x = −rx + b vy − ux + ηx + ξx,

v˙y = −ry − b vx − uy + ηy + ξy, (2.13)

v˙z = −rz − uz + ηz + ξz.

Eqs. (2.9, 2.10, 2.13) are differential, first-order in time and contain only white noises εs and ξs. Hence for the joint probability P ({s, vs, rs, ηs : s = x, y, z}; t), that now contains additional variables rs and ηs, we obtain the Fokker-Planck equation following the standard recipe (see (B.2, B.3) in AppendixB)[38]: X ∂tP = {−vs ∂sP + (rs + us − ηs)∂vs P + ∂rs [(κrs − γκvs)P ] s=x,y,z (2.14) +θ ∂ (η P ) + 1 qθ2 ∂2 P + 1 q ∂2 P } + b(v ∂ − v ∂ )P. ηs s 2 ηs 2 w vs x vy y vx For general (confining, but not necessarily harmonic) potentials u(x), eq. (2.14) is not solvable even in the stationary limit t → ∞, where ∂tP = 0. In this limit, (2.14) is solvable only when FDR is valid globally, e.g. (2.5) holds and also qw = 0. Then (2.14) produces for t → ∞ the Gibbs distribution for the joint probability of velocity v and coordinate x [38, 39, 52]:

2 Z 2 1 − m ( v +u(x)) − m ( v +u(x)) P (v, x) = e 2kBT 2 ,Z = dvdx e 2kBT 2 . (2.15) Z Note that (2.15) does not contain the magnetic ﬁeld, because the latter does not appear in the m v2 energy 2 ( 2 + u(x)) that determines the Gibbs density. This is the Bohr-van Leeuwen theorem [20, 21]; see [22, 23, 24] for recent discussions. Eq. (2.15) is conﬁrmed below.

2.4 Magnitudes of involved parameters Since we are interested in weak magnetic fields (cf. Footnote1), we use in our estimates B ∼ 1 mT, and the room temperature 300 K of the biological environment. Important biological ions (Na+, K+, Ca2+, Mg2+) have roughly the same mass and size. For example, Ca2+ weights 40 gram/mol = 6.7 × 10−23 g, has effective radius ∼ 0.2 nm and the charge Q = 3.2 × 10−19 Cl. Thus the cyclotron QB 3 frequency of the ion is b = m ∼ 5 × 10 1/s. The friction frequency γ in (2.1) can be estimated using Stokes formula (2.7) using the room-temperature water viscosity η = 10−2 g cm/s. The estimate produces 1013 1/s. We shall accept a smaller value γ ∼ 1012 1/s to be consistent with alternative estimates for γ [5]. 2 2 The frequency ω0 from (2.2) can be estimated from the width of the potential: mω0 x ∼ kBT . p 2 11 −1 For hx i ∼ 1nm we have ω0 ∼ 10 s . The noise intensities q and qw will be estimated from q ∼ qw and (2.6, 2.11). In few cases we shall put q → 0 or qw → 0 for illustrative purposes. As argued in section2, the friction memory induced by the water thermal bath holds: 1/κ ≥ 1/γ. A better estimate of this memory time 1/κ goes via the cage-jump model [53], where a given molecule diffuses by jumping from one cage (made by neighbouring molecules) to another. The characteristic time a molecule spends in one cage is the memory time. This time can be estimated ∆/(kBT ) as τ0 e , where ∆ is the energy barrier (including the hydrogen bonding contribution) that −13 2 defines the cage, τ0 ∼ 10 s , and ∆ = 0.2 eV [53]. Recalling kBT = 0.025 eV at room 2 This estimate for τ0 can be obtained via the linear size Rc ' 0.1 nm of the cage and thermal velocity v of p th cage’s walls: τ0 ∼ R/vth = R/ kBT/m, where m is the mass of a water molecule.

5 ∆/(kBT ) −10 temperature, we get for the memory time 1/κ ∼ τ0 e ∼ 10 s [2]. Thus we accept for further estimates the following hierarchy of inverse characteristic times

12 11 10 3 −1 γ ∼ 10 > ω0 ∼ 10 > κ ∼ 10 b ∼ 10 s . (2.16)

3 Solving the Fokker-Planck equation via moments

We turn to exact equations for second-order moments. We multiply (2.14) by a function ψ(xs, vs, rs, ηs) (s = x, y, z) and integrate over all random variables defining the average as hfi = R dV f P , where Q dV ≡ dxsdvsdrsdηs. The probability distribution P nullifies at infinity, since the potential u(x) is confining. Hence, after partial integration we get Z Z dV f ∂α(g P ) = − dV P g∂αf ≡ − hg∂αfi , (3.1) for any functions f, g of random variables, where α is one of the variables of integration. Using (3.1) we get for any function ψ: X ∂t hψi = {hvs∂sψi − h(rs + us − ηs)∂vs ψi − h(κrs − γκvs)∂rs ψi − θ hηs∂ηs ψi s=x,y,z + 1 qθ ∂2 ψ + 1 q ∂2 ψ } − b v ∂ ψ + b hv ∂ ψi . (3.2) 2 ηs 2 w vs x vy y vx

In the stationary state, the moments are time-independent: ∂t hψi = 0. We now determine some moments from imposing a natural symmetry on the system.

3.1 Moments constrained by the symmetry We now assume that the potential is spherically symmetric p u(x, y, z) = u( x2 + y2 + z2). (3.3)

This assumption holds well for ions bound in (next to) proteins [29, 33]. Our system is rotation- symmetric in the (x, y) plane perpendicular to the magnetic ﬁeld, since all the forces in (2.12) (besides the magnetic ﬁeld term) are spherically symmetric. This means that f(Ax,Bx) and f(Ay,By) have same statistics for any function f and random variables Ai,Bi ∈ {s, vs, ηs, rs; s = x, y}. Again from the rotational symmetry, rotating system by 90 degrees, the coordinate transformation is Ax → Ay and Ay → −Ax. Therefore, hf(Ax,By)i = hf(Ay, −Bx)i, and

hAxBx − AyByi = 0 and hAxBy + AyBxi = 0 (3.4) where A, B can be x, vy, ηx, etc. For f(x, y) = xy we have hf(Ax,Ay)i = hf(Ay, −Ax)i implying that components of the same quantity are uncorrelated: hAxAyi = 0; e.g. hxyi = 0 and hvxvyi = 0. These results also follow from (3.2).

3.2 Exact equations for the moments Putting products of various random variables into (3.2) we get system of equations; see Appendix B for details. Here we present the main result from (B.19) assuming that FDR (2.5) holds: q0 1 hLi ≡ hxv − yv i = −b w + 2 × hu (η − r )i , (3.5) y x γκ2 γκ2 x y y 0 qw ≡ qw + huzvzi , (3.6) q q0 b2 + γκ + κ2 b hu (η − r )i + κ hu (r − η )i v2 + v2 = + w + 2 × x y y x x x , (3.7) x y γ γ κ2 γκ2 q + q0 hu (r − η )i hxu + yu i = w + 2 × x x x , (3.8) x y γ γκ

6 where L = xvy − yvx is the z-component of the orbital momentum (i.e. the component along the magnetic field), and where r − η is the excess force acting on the ion; cf. (2.12). It is seen 0 qw from (3.5) that generally hLi 6= 0. Below we confirm that the term −b γκ2 in (3.5) determines the qualitative behavior of hLi and indicates on the diamagnetic response to the external magnetic field: hLi/B < 0, i.e. hLi (and hence the magnetic momentum that is proportional to it) is anti-parallel to the magnetic field. Now hLi nullifies for three cases; see (3.5). First, hLi = 0 when κ → ∞ in (3.5), since now we get that for long times the system holds FDR with white noises and memoryless friction. Hence it relaxes to the Gibbs distribution (2.15) that does not contain the magnetic field b, and leads to hLi = 0. In the same limit κ → ∞ we get from (3.7) the known thermal expression for 2 2 2 vx + vy = 2 vx . Second, hLi = 0 for qw = 0 (zero intensity of the white noise) where FDR 0 again holds. Now we should note that qw = huzvzi = huzi hvzi = 0 in (3.6), because within the (stationary) Gibbs distribution (2.15), the coordinate and momentum factorize and hvi = 0. In (3.8) q/γ is already the result of the Gibbs distribution, as seen from (2.15, 2.6) upon using (2.11). Hence hux(rx − ηx)i = 0. For the same reason hux(ηy − ry)i = 0 in (3.7), which leads to hLi = 0 in (3.5). Third, hLi = 0 for b → 0 (zero magnetic field), this time because the spherical symmetry holds for the stationary state, which e.g. leads to huxηyi = − huxηyi = 0 after inverting x → −x. Below we work out (3.5–3.8) for particular cases.

3.3 Harmonic potential 2 For the harmonic potential (2.2) we get huifji = ω0 hxifji for i, j = x, y, z. The moment equations become linear and are hence exactly solvable; see AppendixB. Instead of (3.5–3.8) we now get explicit expressions: q hLi ≡ 2H = hxv − yv i = −b w (3.9) y x γκ2 q q κ2 + ω2 + γκ + b2 2S ≡ v2 + v2 = + w × 0 (3.10) x y γ γ κ2 2 2 2 2 q qw κ + ω0 2R ≡ x + y = 2 + 2 × 2 , (3.11) γω0 γω0 κ

2 2 where we recall the symmetry features, e.g. H = hxvyi = − hyvxi, S = vx = vy etc. Eq. (3.9) confirms the diamagnetic response. Though the form of this response changes for non-linear potentials, the fact of diamagnetic response stays intact for all confining non-linear potentials we were able to check; see below. Note that hLi in (3.9) is non-zero both due to the non-zero magnetic field and the white noise with intensity qw acting on the ion. The additional factor huzvzi in (3.6) nullified, since for the harmonic potential (2.2) the motion in the z-direction separates from the 2 1 2 D d(z ) E motion in the (x, y)-plane and then huzvzi = 2 ω0 dt = 0 in the stationary state. Since x2 + y2 is the square of the characteristic radius, the (average) angular velocity hΩi that corresponds to hLi in (3.9) is obtained as

2 −1 hxvy − yvxi q κ hΩi = 2 2 = −b 1 + 2 + 1 . (3.12) hx + y i qw ω0

xvy −yvx We confirm in AppendixC that (3.12) is indeed the average of the angular velocity Ω = x2+y2 . q κ2 It is seen from (3.12) that under natural conditions 1 and 2 1 (discussed in section 2.4) qw . ω0 . the angular velocity is of the same order of magnitude as the cyclotron frequency b. Thus the diamagnetic cyclotron motion induced by the magnetic field survives both high friction and high temperature: both effectively cancels out from q in (3.12). qw Thus (3.12) describes ion rotation with radial velocity ∼ 103 s−1; see (2.16). The linear velocity of this motion can be estimated from 10−7 cm × 103 s−1 ∼ 10−4cm/s. By its order of magnitude this (linear velocity) coincides with the ordered motion velocity inside of the cell; see page 24 of

7 Ref. [39]. In contrast to (free) diffusion, such an ordered motion is led by active sources via energy dissipation [39]. AppendixA compares in detail the diamagnetic moment induced by (3.9) with the quantum diamagnetic moment that is produced by electron motion inside of atoms. This quantum effect is responsible for the weak diamagnetism of water [20]. AppendixA concludes that local magnetic moments for both effects can be comparable with each other, though the physical content of these diamagnetic effects is quite different: (3.9) produces an ordered motion with a velocity reasonable for a strongly frictional cell environment, while the quantum diamagnetism relates to an ordered intra-atom motion of electrons that do not feel any friction.

3.3.1 Fluctuations in cyclotron motion For the harmonic potential (2.2) the joint probability density for the coordinates and velocities is Gaussian, as seen e.g. from the fact that (2.12) is a linear equation with Gaussian noises. T Introducing a column vector φ = (vx, vy, x, y) (transposed row vector), we get from (3.9–3.11) for this joint density:

 S 0 0 −H 1 1 T −1  0 SH 0  P (φ) = √ exp − φ Σ φ , Σ =   . (3.13) (2π)2 detΣ 2  0 HR 0  −H 0 0 R

In (3.9, 3.12) we studied the average hLi of the orbital moment L = xvy − yvx and estimated the 2 2 average hΩi of the angular velocity Ω = (xvy − yvx)/(x + y ). Next, we shall study fluctuations of these quantities; details of calculations are presented in AppendixC. For a random variable X, R we define hXiY = dXXP (X|Y ) to be its conditional average for a fixed Y . Starting from the Gaussian joint density (3.13) we confirm (3.12) for hΩi = H/R, and also find [see (C.22, C.13) of AppendixC]:

2 2 1 RS − H p Ω2 − hΩi = , ρ ≡ x2 + y2 (3.14) ρ ρ ρ2 R L2 − hLi2 = 2(RS + H2). (3.15)

Note that the full variance of the angular velocity Ω does not exist because of singularity at p 2 2 ρ = x + y = 0; see (3.14) and AppendixC. However, hΩiρ = hΩi does not depend on ρ. Hence we focus on the conditional dispersion (3.14) at a given radius ρ. We shall now compare the dispersion of Ω with its average, i.e. calculate from (3.12, 3.14) the signal-to-noise ratio SNR:

| hΩi | ρ|H| SNR ≡ q = p (3.16) 2 2 R(RS − H2) hΩ iρ − hΩiρ √ Taking ρ = R and recalling that Larmor’s frequency b is much smaller than all other relevant frequencies, we get from (3.16):

H SNR ' √ . (3.17) RS

−8 Using γ > ω0 > κ from (2.16), we find for a single ion SNR ∼ 10 . Assuming that there are sufficiently large number Nion of (relatively√ independent) ions in identical conditions, the above + estimate for SNR will be multiplied by Nion. In a cell the concentration of potassium (K ) ions is 25 −3 + 25 −3 10 8 × 10 meter [19]. Assuming that bound√ K ions are 10 meter , there are Nion ∼ 10 ions −3 in a cell with size 10 µ m. We get SNR × Nion ∼ 10 for the effective SNR. Such values for SNR are comparable with electric current in a good conductor, i.e. a well-known physical phenomena,

8 where the order motion competes with thermal ﬂuctuations 3. For bound Ca2+ ions the situation is similar, since their concentration is 1024 meter−3.

3.3.2 The average torque balance Let us return to (2.12), vector multiply it by x and average the result

d (x × v) = − hx × ri + hx × ηi + hx × ξi + b hv zi − e hxvi , (3.18) dt z where we simplified the double vector product and omitted the contribution from ux that nullifies due to (3.3). Taking (3.18) to the stationary state (hence omitting averages of time-derivatives, e.g. hxvi = 0), we find the average torque balance in the stationary state:

hx × ri = hx × ηi + hx × ξi + b hv zi . (3.19)

1 D dz2 E Projecting (3.19) to the z-axis and using hvzzi = 2 dt = 0 we ﬁnd that the average torques generated by the friction and noises compensate:

hxry − yrxi = hxηy − yηxi + hxξy − yξxi . (3.20)

To calculate (3.20) for the harmonic potential (2.2), we compare (3.9) with (3.5) that produces 2 2 huxηyi = huxryi. Next, 2 huxηyi = hxηy − yηxi and 2 huxryi = hxry − yrxi. Hence ω0 ω0

hxry − yrxi = hxηy − yηxi , hxξy − yξxi = 0. (3.21)

Note that though the non-zero orbital momentum L is due to the white noise [cf. (3.9)], the latter does not generate any average torque. The following explicit expression reads from AppendixB by analogy to (3.9–3.11):

bqκ2 hxry − yrxi = hxηy − yηxi = 2 2 2 γκ 2 2 . (3.22) b κ + (κ + 2 + ω0)

This holds as well for qw = 0, i.e. in the equilibrium situation. Hence we got an important addition to the Bohr-van Leeuwen theorem: even though the equilibrium probability density P (v, x) in (2.15) does not depend on the magnetic field – and hence the orbital magnetic moment is zero – the average torque exerted by the friction force in the equilibrium state does depend on the magnetic field. Eq. (3.22) nullifies for κ → ∞, where we return to memoryless friction. This can be seen directly from (3.20), where hxry − yrxi → γ hxvy − yvxi = 0 reduces (in the limit κ → ∞) to the orbital momentum, which nullifies at equilibrium. Using (2.6), (2.11) and (2.16) we obtain from (3.22) for the real torque m hxry − yrxi of the −29 4 friction force mr: m hxry − yrxi ∼ 10 Joules . Further studies are needed for understanding the physical relevance of (3.22). For instance, it can be detected if the average balance between the friction force torque hxry − yrxi and the random force torque hxηy − yηxi is broken due to a sudden rise of temperature that will increase the last torque and make it observable. Obviously, such an effect is impossible if hxry − yrxi = hxηy − yηxi = 0 at equilibrium.

3The maximal current density in cooper is 107A/m2 = neu where n = 1029 m−3 is the concentration of conducting electrons in cooper (it coincides with ion concentration in the cell), e = 10−19 Cl is the electron charge, and u is the −3 6 regular velocity of electron. Thus we get u = 10 m/s. Taking the characteristic thermal velocity as vT = 10 m/s −9 (which is predicted from the quantum, Fermi-Dirac thermal√ density [20]), we get SNR = u/vT = 10 . Since the 3 14 14 −2 number of electrons in (10µm) is 10 , the effective SNR is 10 × u/vT ∼ 10 . 4Let us argue that ∼ 10−29 Joules for the torque is not negligibly small. This can be seen via estimating effective electric fields that are necessary for creating such a torque [45]. For an ion moving around 1 nm, the torque of 10−29 Joules implies the effective electric field E found from QE × 1nm = 10−29, where Q ∼ 10−19 Cl is ion’s charge; hence E = 0.1V/m, which is a typical electric field in metals.

9 3.4 Non-linear potentials and cumulant approximation For non-linear potentials, the moment equations generated from (3.2) are not solvable exactly. A direct solution of stationary Fokker-Planck equation (2.14) is also not available. Hence we shall rely on the weakly non-linear (cumulant) approximation in (2.14) and numerical solution of (2.9, 2.10, 2.13). Now consider an isotropic, non-linear potential of the following form: A B U(|x|) = (x2 + y2 + z2) + (x2 + y2 + z2)2,B ≥ 0. (3.23) 2 4 For B = 0 and A > 0 this reduces to the harmonic potential that leads to a Gaussian stationary probability density of the involved variables. The harmonic potential has one stable point: x = y = z = 0. For A < 0 and B > 0 potential (3.23) is still conﬁning, but has a continuum of stable points determined from |x|2 = |A|/B. Let a, b, c and d be zero average Gaussian variables. We get for their fourth-order moments:

habcdi = habi hcdi + haci hbdi + hadi hbci . (3.24)

4 2 2 3 2 For example, x = 3 x or vxx = hvxxi x . The essence of the cumulant approximation is that if B > 0 in (3.23) is small, we can still apply (3.24); e.g.

3 2 2 huxfi =A hxfi + B x f + y xf + z xf (3.25) ≈ A + B 3 x2 + y2 + z2 hxfi ≈ A + 5B x2 hxfi ,

0 and similarly for uy and uz. In this approximation huzvzi ∼ hzvzi = 0 so qw = qw; see (3.6). In (3.25) we also assumed x2 ≈ z2 which is true when magnetic ﬁelds are small. Now deﬁne

2 2 ω0 = A + 5B x . (3.26)

Eqs. (3.25, 3.26) show that within the cumulant approximation, the potential (3.23) is eﬀectively harmonic, and we can use the harmonic solution (3.9–3.11). Now (3.11) is a self-consistent equation for x2 . Note that within the cumulant approximation the orbital momentum is given by its harmonic-potential expression (3.9). Figs.1 and2 compare results of numerical simulation with predictions of the cumulant approximation (the shape of potentials is also shown). We see that the approximation expectantly breaks down for highly nonlinear potentials. We emphasize, however, the cumulant approximation still qualitatively describes the behaviour of the orbital momentum and of x2 in the nonlinear regime. In particular, even for non-linear potentials, the response to the static magnetic ﬁeld is diamagnetic: hLi/b < 0. Fig.3 illustrates a direct check for the validity of (3.24). It can be seen 2 again that approximation x4 ≈ 3 x2 does not work for highly non-linear situations. But for 2 such potential we get x4 < 3 x2 .

4 Paramagnetic response

Eq. (3.9) shows that whenever a white noise influences the ion subject to an equilibrium thermal bath, the magnetic response is diamagnetic. A paramagnetic response is also possible provided that the structure of non-equilibrium is different [25]. In this context, let us recall that within the equilibrium quantum statistical physics, the damagnetic response is typical for moving charges, while spin degrees of freedom respond paramagnetically (i.e. the induced magnetic moment is aligned to the magnetic field) [2]; see AppendixA. In Langevin equation (2.12) we now assume that relation (2.5) is absent, i.e. there is no specific relation between the memory times 1/κ and 1/θ of frictions and noise, respectively [54, 55].

10 theory 1.4

H 1.5

1.6

1.7 4 3 2 1 A 0 1 2 3

Figure 1: The average orbital momentum hLi = 2H in different potentials (3.23) for γ = 0.3, b = 1, κ = 1 and B = 0.05, qw = 1, and q = 0.The points are obtained by solving (2.9, 2.10, 2.13) numerically, the dashed line is the theoretical prediction (3.9) of the cumulant approximation. Insets visualize the shape of the potential at the corresponding value of A; cf. (3.23). More nonlinear potentials result in larger deviations from the cumulant approximation. However, we see that the cumulant approximation (3.9) bounds from above the real orbital momentum. Eqs. (2.9, 2.10, 2.13) are first-order stochastic differential equations. Their numerical integration was done by Eu- ler–Maruyama method with the integration step ∆t = 0.0001. This is a simple generalization of the Euler method for ordinary differential equations to stochastic differential equations [39]; this and more complex methods are reviewed in Ref. [39], page 60. The average and the error were estimated over 1000 sample trajectories in the stationary regime. Errors were estimated via the standard error of the mean; ±3σ error-bars are shown.

25 theory

R 15

4 3 2 1 A 0 1 2 3

Figure 2: The dependence of R = x2 from A [cf. (3.23)] for the same parameters and methods as in Fig.1. The dots are values from numerical simulations. The dashed line is the theoretical prediction. The latter one is the solution of eqs. 3.11, 3.26 and we see that cumulant approximation 3.24 gives qualitatively correct result. Error bars are smaller then the point size.

Restricting ourselves with the harmonic potential (2.2), we ﬁnd from (B.21) of AppendixB:

2 q θ b(κ − θ)(κ + θ) qwb hLi = 2 2 2 2 2 2 − 2 (4.1) γ κ (ω0 + θ + γθ) + b θ γκ where 1/κ is the characteristic memory time of the friction, 1/θ is characteristic correlation time of the fluctuating force. Eq. (4.1) confirms that L = 0 whenever FDR holds globally, e.g. (2.5) is valid, κ = θ, and the white noise is absent qw = 0. For κ = θ we revert from (4.1) to (3.9). For qw → 0, a diamagnetic behaviour is seen from (4.1) for θ > κ [cf. (3.9)], while for θ < κ we get a paramagnetic response. A similar transition between diamagnetic and paramagnetic response was found in [25] for a related model. We emphasize that a paramagnetic response is obtained also for a more complex situation, where θ = κ as in (2.5), qw → 0, but ξ in (2.1) is a colored noise 0 0 θξ −θξ|t−t | with θξ < θ = κ. Here hξ(t)ξ(t )i ∝ δij 2 e is the correlation function of ξ; cd. (2.4). Let us relate hLi= 6 0 in (4.1) with mechanical quantities. For clarity let us focus on the case κ → ∞ and qw → 0, i.e. memoryless friction and no white noise. Here the response is surely paramagnetic; cf. (4.1). Now the non-equilibrium aspect comes with a finite θ, i.e. with the colored noise. From moment equations worked out in AppendixB we obtain (without assuming that the

11 1.0

0.9 x4 3 x2 2 0.8

0.7

0.6 4 2 A 0 2

Figure 3: The cumulant approximation is based on x4 = 3 x2 2. Above we plotted their ratio for the same parameters as in Fig.1. It can be seen that the approximation does not work for highly nonlinear potentials, i.e. for lower values of A. potential is harmonic): 1 1 hLi = hxη − yη i − hxu − yu i . (4.2) γ y x γ y x

The term hxuy − yuxi in (4.2) is the torque generated by the potential force ux. It nulliﬁes since the potential u is assumed to be isotropic. Then hLi 6= 0 in (4.2) is due to the vector product 5 ez(x × η), which is just the torque generated by the colored noise (force) η.

5 The anatomy of orbital magnetism

We saw above that in the stationary state there emerges a magnetic field-induced orbital magnetism (3.9) that relates to the average angular velocity hΩi; cf. (3.12). Now we aim at a more detailed understanding of this phenomenon, e.g. is the stochastic motion that leads to hΩi 6= 0 is similar to real rotations or it is more like a creeping? To understand such questions we need to study autocorrelation functions, e.g. hx(t0)x(t0 + t) + y(t0)y(t0 + t)i = hx(0)x(t) + y(0)y(t)i that in the stationary state depends on the time-difference t only. Since the Fokker-Plank equation (2.14) does not account for autocorrelation functions, we will use the Fourier method [21] for solving Langevin’s equation (2.12) directly 6. To make the solution more transparent, we restrict ourselves with the harmonic potential (2.2) and consider only the white noise, i.e. q = 0 and qw > 0 in (2.12). These restrictions suffice for having a well-defined average angular motion induced by the magnetic field; cf. (3.9). Hence we focus on x and y coordinates of (2.1) that read together

ζ ≡ x + iy, ξ ≡ ξx + iξy, (5.1) Z t ¨ 2 ˙ 0 −κ|t−t0| ˙ 0 ζ(t) = ξ(t) − ω0 ζ − ibζ(t) − γ dt κ e ζ(t ), (5.2) −∞ where in (2.1) we took t0 = −∞ aiming to focus on the stationary state. Eq. (5.2) belongs to the class of linear oscillators driven subject to non-local friction and noise. Such systems were studied in [24, 25, 27, 40, 41, 42, 43, 44, 54, 55, 56, 57]. We apply Fourier’s transform to ζ and ξ in (5.2): Z Z dt X(t) = dω eiωtX˜(ω), X˜(ω) = e−iωtX(t), (5.3) 2π 5One can ask whether the diamagnetic orbital momentum L in (3.9) also relates to the average torque. Not quite so: focusing on the case q = 0 (i.e. the colored noise η in (2.12) is absent) we ﬁnd for hLi in (3.9) from moment 1 1 equations worked out in AppendixB: hLi = γκ hrxvy − ryvxi + γ hxuy − yuxi. The term hxuy − yuxi nulliﬁes as for (4.2). Then hLi 6= 0 in (3.9) is due to the vector product ez(r × v), which can be called a generalized torque of the friction force r. 6The Fourier solution is partially numeric, and it does not recover exact results of the Fokker-Planck equation. It is also not useful for approximate treatments of anharmonic potentials. Hence we employ it for autocorrelation functions only.

12 1000 theoretical mean sample mean 500 ) d a r (

l 0 g n A 500

1000

0 5 10 15 20 25 30 Time -4.5 turns 4.6 turns 4.6 turns 4.4 turns

2 0 2 2 0 2 2 0 2 2 0 2

Figure 4: Stochastic trajectories for the two-dimensional harmonic oscillator (2.2) obtained from solving the x and y components of (2.9, 2.10, 2.13) numerically via the Euler-Maruyama method; see the caption of Fig.1. Top row: the angle Ω t around the origin for 20 diﬀerent trajectory realisations. Each color refers to a separate realisation. The thick blue (dot-dashed) line is the average over all 20 realisations and the thick black (dashed) line shows the theoretical result for hΩi t; see (3.12). Second row: 4 random trajectories in the (x, y) plane; the red point p 2 p 2 denotes the origin (0, 0). The parameters are b = 1, γ = 100, κ = 10, ω0 = 90, qw = 1, q = 0, σ ≡ hx i = hy i; see (3.11). Hence the average angular velocity is hΩi = −0.99 according to (3.12). The relaxation times are 0−1 0−1 0−1 00−1 00−1 00−1 (u1 , u2 , u3 ) = (0.1122, 1.8454, 1.8265) and the corresponding rotational periods are (u1 , u2 , u3 ) = (−909.1, 0.0104, −0.0105); cf. (5.8). Thus for t > 3 the system is in the stationary state, but rotations are well- 00−1 0−1 00 −1 0−1 00 00 visible, since u2 u2 and |u3 | u3 in (5.9). The diﬀerence between u2 and |u3 | is due to b 6= 0. For stochastic trajectories, the darker points refer to more recent times, brighter points are earlier times.

100 theoretical mean sample mean 50 ) d a r (

e l 0 g n A 50

100 0 5 10 15 20 25 30 Time 0.1 turns -1.8 turns -0.1 turns -1.1 turns

2 0 2 2 0 2 2 0 2 2 0 2

Figure 5: The same as in Fig.4, but for ω0 = 10. Hence hΩi = −0.50 in (3.12). The relaxation times and rotation 0−1 0−1 0−1 00−1 00−1 00−1 periods are (resp.): (u1 , u2 , u3 ) = (1.0917, 0.2236, 0.2168) and (u1 , u2 , u3 ) = (−129.9, 0.030, −0.031). 0−1 0−1 00−1 0−1 The relaxation time u2 ' u3 is still some 10 times larger than the rotation period u2 ' |u3 |. where X(t) = (ζ(t), ξ(t)) and X˜(ω) = (ζ˜(ω), ξ˜(ω)), and get from (5.2): ˜ ˜ ξ(ω) ζ(ω) = γ κ 2 . (5.4) iω(iω + κ+iω + ib) + ω0

13 D E ∗ ˜ ˜∗ qw Now (2.4, 5.1) lead to hξ(t1)ξ (t2)i = 2qwδ(t1 − t2). This implies ξ(ω1)ξ (ω2) = π δ(ω1 − ω2) from (5.3). Hence we ﬁnd from (5.4): Z D E ∗ iω1t1−iω2t2 ˜ ˜∗ hζ(t1)ζ (t2)i = dω1 dω2 e ζ(ω1)ζ (ω2) (5.5) q Z |ω − iκ|2 w iω(t1−t2) = dω e 2 2 2 . (5.6) π |(ω − iκ)(ω + ωb − ω0) − γκω|

For t1 − t2 > 0 only the poles with positive imaginary parts contribute into (5.6), since the integration contour can be closed above the real axis (in total, (5.6) has 6 simple poles). It turns out (as conﬁrmed in AppendixD and discussed below) that those poles ω1, ω2 and ω3 are the roots of a cubic equation

2 2 (ω − iκ)(ω + ωb − ω0) − γκω = 0. (5.7) Eq. (5.7) reads after introducing ω ≡ iu:

3 2 2 2 u − (κ + ib) u + u(γκ + ω0 + ibκ) − ω0κ = 0. (5.8) Now all roots of (5.8) have positive real parts. This is shown via the Routh-Hurwitz criterion; see AppendixD for details. Hence via the residue theorem we get from (5.6)

3 3 0 00 ∗ X iωnt X −u t−iu t hζ(t1)ζ (t2)i = Cne = Cne n n , (5.9) n=1 n=1 0 00 where we introduced the real and imaginary parts un = un + i un for the roots of (5.8), and 0 where Cn are t-independent pre-factors we shall not be interested in. Thus un > 0 correspond to 00 00−1 inverse relaxation times, while un refer to rotation frequencies. We also refer to un as rotational period, leaving out 2π factor. We have rotational (underdamped) motion if at least one condition u0 u00 holds. Otherwise, the motion is relaxational (overdamped). It should be clear from n n D E ˙ ˙∗ the above derivation that the autocorrelation functions of other quantities, e.g. ζ(t1)ζ (t2) , are expressed as in (5.9), but with different pre-factors. Consider a particular case in (5.8). For κ 1, i.e. in the memoryless friction regime, (5.8) 0 00 0 00 0 00 leads to u1 ' κ u1 , u2, u2 , u3, u3 . Hence the the contribution from n = 1 quickly disappears 2 2 from (5.9), and we are left with two roots of the quadratic equation u − u(γ + ib) − ω0 = 0 found from (5.8). For a small b, these two roots predict that the transition from overdamping to underdamping takes place at γ ∼ 2ω0. For not so large κ the underdamping is facilitated, since the friction γ in (5.8) is multiplied by κ. Figs.4–7 show realisations of random trajectories from (5.2) both projected to the angle variable (first row) and in the (x, y)-plane (second row)for various ranges of parameters. In the figures the unit of time is taken to be 1. Hence the values of parameters do not coincide with their realistic estimates from section 2.4. But ratios of parameters are kept realistic, except for the magnitude of the cyclotron frequency b that is taken larger than its natural value for making visible the effects of the magnetic field. Parameters of Fig.4 are those of underdamping due to memory: the relaxation times are more 00−1 0−1 00 −1 0−1 than 100 times larger than the rotation periods: u2 u2 and |u3 | u3 in (5.9). Hence there are almost deterministic rotational realisations with small stochastic effects; see the second (bottom) row of Fig.4. The first (top) row of Fig.4 visualizes motion of the ion for longer time intervals, each color for a separate realisation. The vertical axis shows the angle rotated around the origin (0, 0), and a constant slope means constant angular velocity Ω. The slope is constant for 0−1 times smaller than the relaxation time u3 ' 2. Interestingly, for certain realizations, the slope can be constant over longer times. As shown in the first row of Fig.4, ions rotate both clockwise and counterclockwise. But the angular velocities are not the same and on average the ion will rotate clockwise due to the magnetic field, as predicted by (3.12). The empiric average (thick blue line) deviates from the theoretical average (black line) due to a small number of samples.

14 150 theoretical mean sample mean 100 ) d

a 50 r (

e l

g 0 n A 50

100

0.0 0.2 0.4 0.6 0.8 1.0 Time 3.6 turns -1.8 turns 3.0 turns -2.8 turns

2 0 2 2 0 2 2 0 2 2 0 2

Figure 6: The same as in Fig.4, but with κ = 1000 and ω0 = 1000; hence Ω = −0.50 in (3.12). The relax- 0−1 0−1 0−1 00−1 00−1 00−1 ation times and rotation periods are (resp.): (u1 , u2 , u3 ) = (0.001, 0.040, 0.040) and (u1 , u2 , u3 ) = 00−1 (−40.16, 0.00097, −0.00097). The shortest rotation periods are much smaller than the relaxation times: u2 ' 00 −1 0−1 0−1 |u3 | u2 ' u3 .

40 theoretical mean sample mean 20 ) d a r (

l 0 g n A

0 5 10 15 20 25 30 Time -0.5 turns 0.4 turns -0.2 turns -1.2 turns

2 0 2 2 0 2 2 0 2 2 0 2

Figure 7: The same as in Figs.4 and5, but for parameters: κ = 1000, ω0 = 20. The average angular speed induced by the magnetic ﬁeld is small: hΩi = −4 · 10−4; cf. (3.12). With these parameters we are in the regime of (the ordinary) overdamped Brownian motion with a weak memory and a weak magnetic ﬁeld. The relaxation 0−1 0−1 0−1 00−1 00−1 00−1 times and rotational frequencies are (resp.) (u1 , u2 , u3 ) = (0.0011, 0.00922, 0.2407) and (u1 , u2 , u3 ) = (−6.883, 0.840, −22.27). The relaxation times are much shorter than the rotation periods.

Fig.5 shows a moderately damped situation, since ω0 is 9 times smaller than in Fig.4. Rotations are still visible. In contrast to Fig.4, the trajectories are more irregular; see the first row of Fig.5. 0−1 Here the longest relaxation time u1 corresponds to the component with the longest rotational 00−1 period u1 . Indications of this are seen in the first row of Fig.5: some slopes are "flat" with zero rotation on average. Fig.6 demonstrates a memoryless underdamped situation: now both κ and ω are larger than γ, and certain relaxation times are larger than the rotation periods. Both

15 Fig.6 and Fig.4 demonstrate rotational motion, but the underlying mechanisms are different: in the former case it is due to a large ω0 (i.e. strongly confining potential), while in the latter case it is due to memory in friction. Note that trajectories in the second row of Fig.4 are more regular than those in Fig.6. Finally, Fig.7 shows a memoryless situation (i.e. a large κ) and loosely confining potential (moderate ω0). For this case (5.8) has 3 real roots under b → 0: the large rotation periods is due to the small perturbation of the magnetic field. Hence no deterministic rotation is visible. The trajectories in the bottom row of Fig.7 resemble those of the ordinary random walk.

6 Summary

A static magnetic field is not screened by conducting biological matter (in contrast to electric field) and hence penetrates into an organism without changing its shape and magnitude. This is one reason why magnetic field studied is applied in biology and medicine. Magnetobiological experiments motivate us to ask how a weak magnetic field can influence ions in a warm and wet cellular environment, i.e. at room temperatures and high friction [7, 58, 30, 29,8,9]. We propose a model for that influence that follows the ideas of Brownian motion in a magnetic field, but with two additional mechanisms [44, 25]. First, the friction acting on a cellular ion cannot be given by the ordinary memoryless (Ohmic) expression, since the thermal bath for ions is made of water molecules; see section 2.2. Once the (equilibrium bath-generated) friction and noise relate via the Fluctuation Dissipation Relation (FDR, cf. (2.5)), they both have a characteristic memory time, which is estimated in section 2.4. Second, besides the equilibrium noise coming from water thermal bath, the cellular environment has various other noises that have non-equilibrium origin and are expected to be relevant for objects of sub-cellulalr size < 1µ m [36, 37]. Thus such noises do not hold FDR. Here we offered the simplest realizations of above two mechanisms via the Langevin equation with exponential memory kernels; cf. (2.1, 2.4, 2.12). Our main results are as follows: • A weak static magnetic field induces angular motion with (average) cyclotron frequency; see section 3.3. This is despite of room temperatures and high friction of the water thermal bath. The linear velocity that corresponds to this cyclotron motion has the same order of magnitude as other ordered motions in the cell. • The magnetism relates with a confined motion in a confining isotropic potential, but is otherwise not qualitatively susceptible to the form of this potential; see section 3.4. • More detailed understanding of the orbital magnetism is achieved via stochastic trajectories and autocorrelation functions; see section5. The following interesting observations done in this work will be explored elsewhere. – Though the orbital magnetism disappears in the equilibrium state (i.e. when the non- equilibrium noise has the same structure as the equilibrium one), this does not mean that all signs of the magnetic field disappear from that state. In particular, the magnetic field leads to a non-zero average torque by the stochastic force, which–in the equilibrium situation–is precisely compensated by the average torque generated by the friction force; see section 3.3.2. While this precise compensation precludes the straightforward observation of the average torque, it might still be observed indirectly. – Since the prediction for the cyclotron motion of cellular ions relates to non-equilibrium, the non-reproducibility of certain magnetobiological experiments can be possibly explained via meta- stability (fragility) of non-equilibrium states. Once there is a possibility for a cyclotron motion in a high-friction and high-temperature cellular environment, future studies should devote attention to time-dependent electromagnetic fields and possible resonance phenomena. Another pertinent topic is to incorporate electrostatic interactions between the ions that are neglected here. Further research can establish relations of our results with active matter under magnetic field [43], electromagnetic noise [23, 45], and nanomagnetic particles [59].

16 Acknowledgements We thank David Petrosyan and Gurgen Adamian for discussions. AM is supported by Cambridge Trust Scholarship. AA is supported by SCS of Armenia, grant 20TTAT-QTa003.

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Appendices

A Comparing the non-equilibrium classical and equilibrium quantum average magnetic moments A.1 Comparison with quantum noise In the context of (2.1, 2.5), let us emphasize that the very possibility of applying any classical Langevin description demands that the characteristic (correlation) of the noise is smaller than the quantum correlation time kBT/~ [52]. For (2.1) this ﬁrst of all implies

κ kBT/~, (A.1) because ~/(kBT ) is the characteristic correlation time of the quantum noise [52]. Moreover, the difference in (A.1) should be sufficiently large, so that the correlation frequency of ξ(t) in (2.1), which is was assumed to be much larger than κ (i.e. effectively white), also holds (A.1). In (2.16) 10 13 we estimated κ ∼ 10 1/s, while kBT/~ ∼ 10 1/s. The orbital magnetism studied in section3 is a non-equilibrium classical phenomenon. We want to compare it with magnetic effects predicted by equilibrium quantum physics [20, 21, 60]. There are at least two types of such effects: paramagnetism of nuclear spins and diamagnetism of atoms and molecules that is due to the quantum cyclotron motion of electrons bound in atoms.

A.2 Comparison with quantum paramagnetism Nuclear spins have a tiny equilibrium paramagnetic average magnetic moment even at room temperature [20, 60]. This polarization is experimentally visible at magnetic fields B & 1T due to a sufficiently large number of identical nuclei. Hence it is interesting to compare the magnitude of this quantum equilibrium effect with the non-equilibrium magnetic moment Mc = QL/2 generated according to (3.9): k T Qb QB M = − B , b = (A.2) c mκ2 m

20 where Q is the elementary charge, b is the cyclotron frequency and we used (2.6) in (3.9) as a sensible estimate for qw ∼ q; recall (2.11) in this context. Note that the sign of the magnetic moment does not depend on the sign of Q. 1 The interaction energy of the nuclear spin 2 with the magnetic ﬁeld Bez (directed along the 2 z-axis, ez = 1) is [20, 60]

−~BQsˆz/(2m), (A.3) 1 where sˆz is the z-component of the spin- 2 operator (third Pauli’s matrix), and m is the nucleus mass, which is taken to be equal to ion’s mass in (2.1) (they anyhow have the same order of magnitude). The g-factor in (A.3) is put to 1, since we aim at qualitative estimates. Hence ˆ M = ~Qsˆz/(2m) is the magnetic moment operator [60]. Its equilibrium average value is D E Q QB 2Qb Mˆ = ~ tanh ~ = ~ , (A.4) 2m 4mkBT 8mkBT h i where we employed tanh ~QB ' ~QB , which holds due to high (room) temperatures. 4mkBT 4mkBT We now get from (A.2, A.4): 2 |Mc| 8 kBT D E = 2 , (A.5) Mˆ κ ~

D E i.e. if (A.1) holds say by three order of magnitude, then |Mc| exceeds Mˆ by (at least) six orders of magnitude.

A.3 Comparison with quantum diamagnetism Turning to quantum diamagnetism, let us quote the standard Langevin’s formula for the average magnetic moment of a molecule (atom) that is created by the external magnetic ﬁeld via induced cyclotron motion of its electrons [20, 21]:

2 2 ZQ ri B hMid = − , (A.6) 6me where ri is molecule’s radius, Q is the electron charge, me is the electron mass, and Z is the number of electrons in the molecule. Note that albeit (A.6) is a quantum eﬀect (we recall that classically there is no equilibrium magnetism) it does not contain ~ explicitly [20, 21]. It also does not contain any direct reference to the temperature, since it refers to the ground state motion of electrons inside of the molecule. This motion does not feel the temperature [20, 21]. Let us now see that (A.6) correctly reproduces the diamagnetic volume magnetic susceptibility of water (i.e. the main substance of any cell). To this end, note that (A.6) leads to the macroscopic magnetic moment χdB/µ0 [20], where

2 2 Nwµ0ZQ ri χd = − , (A.7) 6me −6 is the sought susceptibility. Here we denoted µ0 = 1.3 × 10 and Nw is the number of water 3 1 29 −3 molecules in meter . In normal conditions Nw = 3 × 10 meter , (this comes from the water 3 −10 −19 density 1g/cm and molar mass water 18 g/mol) ri = 2 × 10 meter, Q = 1.6 × 10 Cl, Z = 10, −31 and me = 9 × 10 kg. Putting these numbers into (A.7) we obtain:

−5 χd = −7.4 × 10 , (A.8) which is close but still larger by its absolute value than the standard value −9 × 10−6 for water [2]. This discrepancy can be explained by noting that there is a (positive!) paramagnetic contribution to be added to (A.8). Overall, the water diamagnetism is explained by (A.6).

21 We compare (A.2) with (A.6):

Mc 6 kBT me D E = 2 2 2 . (A.9) Mˆ Z m ri κ d

−8 Putting into (A.9) standard estimates for the ion (room temperature, ri = 2 × 10 cm, m = −23 −28 10 −1 3 × 10 g, me = 9 × 10 g, κ ∼ 10 s , Z ∼ 10), we ﬁnd that

Mc 6 D E ∼ , (A.10) Mˆ Z d D E i.e. Mc and Mˆ have the same order of magnitude. The macroscopic magnetic moment is larger d for water, since its concentration is larger. Indeed, the concentration of e.g. bound potassium (K+) ions in a cell is roughly 1025 meter−3 [19], i.e. some 4 orders of magnitude smaller than the water concentration. D E We should however recall that the physical meaning of Mc is different from that of Mˆ . The d DlatterE refers to electron motion inside of atoms. The electron motion is frictionless and noiseless: Mˆ refers to the ground state of an atom with its electrons subject to a self-consistent field d [20, 21]. In contrast, Mc refers to ion rotation on scales & 1 nm and subject to both strong friction and room-temperature thermal noise; see (2.16). D E Finally, let us mention that besides frictionless and noiseless Mˆ , there is a theoretical d possibility of a dissipative equilibrium quantum diamagnetism for ion’s motion. It is described by the quantum Langevin equation with a local friction and with the quantum noise with correlation time (A.1)[46]. This effect is however suppressed by friction [46], i.e. it is small under the strong cellular friction (2.16).

B Equations of moments

First we recall the recipe of writing down the Fokker-Plank equation from the Langevin equation. Consider the following white-noise Langevin equations (assuming summation over repeated indices): d ξ = h (ξ, t) + g Γ (t), h Γ (t)Γ (t0) i = δ δ(t − t0). (B.1) dt i i ij j i j ij Then the corresponding Fokker-Plank equation reads [38, 39]

2 ∂ ∂ (1) ∂ (2) P = − Di (x, t) P + Dij P , (B.2) ∂t ∂xi ∂xi∂xj 1 D(2) = g g ,D(1)(x, t) = h (x, t). (B.3) ij 2 ik jk i i Using (B.2, B.3) we get (2.14) from (2.9, 2.10, 2.13). Turning to moments generated via (3.2), let us recall that many moments are equal or cancel each other; see section 3.1. Hence we denote:

hAx,Byi ≡ hAxBy ± AyBxi , ± (B.4) hAx,Bxi± ≡ hAxBx ± AyByi ,

2 2 where e.g. hx, vyi− ≡ hxvy − yvxi or hx, xi+ = x + y . Using a function ψ = g(x, y, z) in the stationary regime of (3.2) we ﬁnd

0 = hvx∂xgi + hvy∂ygi + hvz∂zgi , (B.5)

22 hence g = xy : hxvyi + hyvxi = 0 2 2 2 g = x ; y ; z : hxvxi = hyvyi = hzvzi = 0 (B.6)

g = u(x, y, z): hvxux + vyuy + vzuzi = 0 One can easily show that ﬁrst-order moments, i.e. average values of the random variables are all 2 2 zero. Then consider ψ = ηxηy, ηx, ηy in (3.2):

2 2 1 ηx = ηy = 2 qθ, hηxηyi = 0. (B.7)

Another set of simple relations is found by putting ψ = ηij and ψ = ηirj (i, j = x, y) into (3.2):

hviηji = θ hxiηji , (B.8) κθ hr η i = γµ hx η i , µ = . (B.9) i j i j κ + θ

2 2 Similar relations we get from ψ = rx, ry, xrx, yry and xry + yrx:

2 hvirii = κ hxirii , hvx, ryi+ = κ hx, ryi+ , ri = γκ hxirii . (B.10) In the following formulas the left column denotes the function ψ to be employed in (3.2), while the result are reported on the right using notations (B.4):

2 vxηx + vyηy :(θ + γµ) hx, ηxi+ + hux, ηxi+ + bθ hx, ηyi− = θ q (B.11) 2 vyηx − vxηy :(θ + γµ) hx, ηyi− + hux, ηyi− − bθ hx, ηxi+ = 0 (B.12) 2 2 vx + vy : κ hx, rxi+ = θ hx, ηxi+ + (huzvzi + qw) (B.13)

xvy − yvx : hx, ryi− = hx, ηyi− (B.14)

vxry − vyrx : γµ hx, ryi− + hux, ryi− + κ hvx, ryi− = bκ hx, rxi+ , (B.15) where in (B.15) we used (B.9). This system of equations is closed (assuming expressions with u are known). Note that hx, ηxi+ and hx, ηyi− are interesting quantities that appear in intermediate steps of the solution: they refer to (resp.) power and torque of the noise. Likewise, hx, rxi+ and hx, ryi− refers to the friction force. The next three equations contain the quantities that we are interested in:

xry − yrx : γκ hx, vyi− = − hvx, ryi− + κ hx, ryi− (B.16)

rxvx + ryvy : γκ hvx, vxi+ = (γ + κ)κ hx, rxi+ + hux, rxi+ (B.17)

−γµ hx, ηxi+ + b hvx, ryi−

xvx + yvy : hx, uxi+ = hvx, vxi+ + b hx, vyi− (B.18)

+ hx, ηxi+ − hx, rxi+ Now we assume θ = κ, i.e. FDR (2.5) holds. Eqs. (B.11–B.18) are solved as

q0 1 hx, v i = −b w + hu , η − r i y − γκ2 γκ2 x y y − 0 2 2 q q b + γκ + κ b hux, ηy − ryi + κ hux, rx − ηxi hv , v i = + w + − + (B.19) x x + γ γ κ2 γκ2 0 q + q hux, rx − ηxi hx, u i = w + + , q0 ≡ q + hu v i . x + γ γκ w w z z This is the farthermost point where we can reach with a general spherical symmetric potential.

23 B.1 Harmonic potential In the case of harmonic potential

ω2 u = 0 x2 + y2 + z2 (B.20) 2 we can relax the condition κ = θ and still get a tractable solution. The moments that contain 2 potential term become huifji = ω0 hxifji for i, j = x, y, z. After this replacement, (B.11–B.18) becomes a system of linear equations, which is solved as

2 2 2 2 2 2 2 2 2 2 θ (ω0 + θ + γµ)(ω0 + κ + γµ) + b θ κ + ω0 + γ0κ + b 2γ S = q 2 2 2 2 2 2 + qw 2 , κ (ω0 + θ + γµ) + b θ κ 2 2 2 2 2 κ2 2 2 2 2 θ (ω + θ + γµ)(ω + κ + γµ 2 ) + b κ ω + κ 2 0 0 θ 0 (B.21) 2γω0 R = q 2 2 2 2 2 2 + qw 2 , κ (ω0 + θ + γµ) + b θ κ θ2 b(κ − θ)(κ + θ) b 2γ H = q 2 2 2 2 2 2 − qw 2 , κ (ω0 + θ + γµ) + b θ κ where R, S, and H are deﬁned in (3.11).

C Variances of orbital momentum and angular velocity

We start from (3.13):

 S 0 0 −H R 0 0 H  0 SH 0  −1  0 R −H 0  Σ =   , Λ = Σ = β   , (C.1)  0 HR 0   0 −HS 0  −H 0 0 R H 0 0 S where β ≡ (SR − H2)−1. Then probability density (3.13) reads

1 P (φ) ∝ exp − φ>Λ φ (C.2) 2 1 = exp − βR(v2 + v2) + βS(x2 + y2) + βH(xv − yv ) (C.3) 2 x y y x

From (C.3) we calculate the marginal density P (x, y) and conditional density P (vx, vy | x, y):

x2 + y2 P (x, y) ∝ exp − (C.4) 2R P (φ) βR P (v , v | x, y) = ∝ exp − v2 + v2 + βH(xv − yv ) (C.5) x y P (x, y) 2 x y y x " # βR Hy 2 βR Hx2 ∝ exp − v + − v − . (C.6) 2 x R 2 y R

These expressions will be used below.

24 C.1 Variance of the orbital momentum The orbital momentum along z direction is

L = xvy − yvx. (C.9)

To calculate its variance we need E[L2]:

2 2 2 2 2 E[L ] = x E[vy|x, y] + y E[vx|x, y] − 2xy E[vxvy|x, y] (C.10) H2 1 = (x4 + y4 + 2x2y2) + (x2 + y2) (C.11) R2 βR = 8H2 + 2(RS − H2) = 2RS + 6H2, (C.12)

2 where we used x4 = 3 x2 . We ﬁnally get together with the mean of angular velocity:

hLi = 2H, Var[L] = hL2i − hLi2 = 2(RS + H2), (C.13) where we used notations (3.9–3.11).

C.2 Mean and variance of the angular velocity Ω Starting from the orbital momentum (C.9) we deﬁne the angular velocity in the (x, y)-plane: xv − yv Ω = y x , (C.14) x2 + y2 and look at its average over (C.3):

xv − yv [Ω] = [ [Ω | x, y]] = y x | x, y (C.15) E E E E E x2 + y2 x y = [v | x, y] − [v | x, y] (C.16) E x2 + y2 E y x2 + y2 E x x Hx y Hy H = + = (C.17) E x2 + y2 R x2 + y2 R R

Note that Ω has a singularity at ρ ≡ x2 + y2 = 0. Thus we study the statistics of Ω at a ﬁxed ρ. From (C.17) it is easy to see that E[Ω | ρ] = E[Ω]. Next, we calculate

" 2 2 2 2 # 2 x vy + y vx − 2xyvxvy E[Ω | ρ] = E ρ (C.18) (x2 + y2)2

" 2 2 2 2 # x E[vy|x, y] y E[vx|x, y] 2xy E[vxvy|x, y] = E + − ρ (C.19) (x2 + y2)2 (x2 + y2)2 (x2 + y2)2

" 2 2 4 4 2 2 2 2 # x + y 1 x + y H 2x y H = E + + ρ (C.20) (x2 + y2)2 βR (x2 + y2)2 R (x2 + y2)2 R RS − H2 1 H2 = + , (C.21) R ρ2 R2 and thus H 1 RS − H2 hΩi = , Var[Ω | ρ] = (C.22) R ρ2 R Recalling (C.3), we see that hΩ2i does not exist due to a logarithmic divergence induced by the 1 factor ρ2 .

25 D Routh-Hurwitz criterion in application to roots of Eq. (5.8)

To show that all roots of (5.8) have positive real parts is the same as to show that all roots of

3 2 2 2 z + (κ + ib) z + z(γκ + ω0 + ibκ) + ω0κ = 0, (D.1) have negative real parts. If the coefficients of (D.1) were real, we could apply the Routh-Hurwitz criterion (i.e. necessary and sufficient condition) for the negativity of real parts [61, 62]. If we multiply (D.1) by its counterpart with complex conjugate coefficients, we shall get a six-order 6 polynomial equation with real coefficients {bl}l=1:

3 2 2 2 3 2 2 2 [z + (κ + ib) z + z(γκ + ω0 + ibκ) + ω0κ] × [z + (κ − ib) z + z(γκ + ω0 − ibκ) + ω0κ] 6 6 X 6−l ∗ = z + blz = 0, bl = bl = 0. (D.2) l=1

Obviously, the roots of (D.2) have negative real parts if and only if the roots of (D.1) have negative real parts [63]. Hence we can apply the Routh-Hurwitz criterion to the sixth-order polynomial (D.2) whose coeﬃcients are real. To this end deﬁne [61, 62]

hij = 0, if 2i − j < 0 (D.3) = 1, if 2i − j = 0 (D.4)

= b2i−j, if 2i − j > 0, (D.5)

α where i = 1, ..., 6 and j = 1, ..., 6. Next deﬁne six matrices: ∆α = {hij}i,j=1, α = 1, ..., 6. The Routh-Hurwitz criterion tells that all roots of (D.2) have negative real parts if and only if [61, 62]:

det[∆α] > 0, α = 1, ..., 6. (D.6)

It is straightforward to check starting from (D.1) that inequalities in (D.6) indeed hold for γ > 0 and κ > 0. The similar check for b = 0 in (D.1) is easier, since it involves only checking the sign of 3 determinants.