Entrainment of a driven oscillator as a dynamical

PHYSICAL REVIEW E 84, 061134 (2011)

Entrainment of a driven oscillator as a dynamical phase transition Alex Dickson, S. M. Ali Tabei, and Aaron R. Dinner* James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA (Received 28 July 2011; published 20 December 2011) Large deviation theory has emerged as the natural language with which to study transitions between dynamical phases, such as the glass transition. Here we use this approach to show that the entrainment of an oscillator to an external driving force can be described as a phase transition between two regions in a joint space-time representation. Specifically, we numerically obtain exact solutions of the large deviation function for a discrete, finite model of an oscillator under a periodically varying external force. We find that the first derivative of the expectation value of the current diverges in the limit of large system size. For weak forcing, perturbation theory allows us to relate the observed frequency of the oscillator to the spectrum of eigenvalues for the unforced system. We find that in the entrainment region where the oscillator exhibits the frequency of the driving force, there is a strong coupling between the ground state and excited states. DOI: 10.1103/PhysRevE.84.061134

PACS number(s): 05.70.Ln, 05.45.Xt, 05.70.Fh

I. INTRODUCTION

The entrainment of an oscillating system to an external force is a phenomenon that is studied in a broad range of scientific disciplines [1]. A few experimental examples include entrainment in electric circuits [2], excitable glow-discharge plasmas [3], and chemical reactions [4]. Entrainment is also pervasive in living systems on scales ranging from individual cells (where oscillations in cell division can be entrained using periodic expression of cyclins [5]) to whole organisms (where circadian sleep cycles couple to cycles of light and dark [6]). In all of these cases, there is an oscillator that is driven by a periodic external force, and this can exhibit two different dynamical behaviors. If the natural frequency of the oscillator (0 ) is close to the frequency of the driving force (f ), the oscillator can entrain (i.e., exhibit an oscillatory frequency that is the same as the driving force). Alternatively, if f 0 or f 0 , then the oscillator will not entrain to the driving force and will exhibit an oscillatory frequency close to 0 . The window for which entrainment occurs depends on the strength of the driving force, and a plot of this dependence is called the Arnold tongue [1]. How can we understand the transitions into and out of entrainment when the oscillator is stochastic? It is natural to turn to statistical mechanics, but equilibrium approaches based on ensembles in configuration space are ill-equipped to describe the transition between two dynamical phases. Recently however, large deviation theory [7,8] has emerged as the natural language to describe such transitions (e.g., free-flowing versus jammed states of exclusion processes [9] or active versus inactive phases of glassy systems [10–13]). The theory introduces a field s, which is coupled to the observable of interest; then, as in equilibrium phase transitions, the signature of a dynamical phase transition is a discontinuity in either the expectation value of the extensive observable or its derivative as a function of the intensive variable. Here we consider a simple model of an oscillator that allows for direct calculation of the large deviation function: a single particle that hops from site to site on a ring but is biased in

*

[email protected]

1539-3755/2011/84(6)/061134(8)

one direction and hence rotates with a natural frequency. A periodic driving force is introduced that couples to the particle dynamics by modifying the probabilities of moving left and right; this driving force has the ability to entrain the rotation of the oscillator if its frequency is close to the natural frequency of the oscillator. We additionally introduce an intensive field s that couples to an observable that measures the rotational frequency of the particle (A). In this way, even for frequencies of the driving force that are far away from the entrainment window, the s field can drive the particle into entrainment. By obtaining A as a function of s and f , we construct a two-dimensional dynamical phase diagram that shows three dynamical phases: below entrainment ( < f ), entrainment ( = f ), and above entrainment ( > f ). We then use perturbation theory to derive analytical insights into these dynamics and show how the external force and the intensive field modulate the eigenstates of the system and their coupling. In particular, entrainment requires that excited eigenstates commensurate with the frequency of the external force be accessible by intrinsic fluctuations from the ground state in analogy to quantum phase transitions.

II. THE MODEL

Here we construct a simple lattice model of a driven oscillator in which a single particle hops from site to site with a discrete time step (Fig. 1). The dynamics of this particle are biased by an inherent tendency to hop clockwise and an external driving force that is periodic in time P ({l,t} → {l ,t + 1}) ⎧1 r − 2 + γ + (l,t) ⎪ 2 ⎪ ⎪ ⎨ 1 − r − γ − (l,t) = 2 2 ⎪ r ⎪ ⎪ ⎩ 0

if l = l + 1, if l = l − 1, if l = l, otherwise,

(1)

where P ({l,t} → {l ,t + 1}) is the probability of hopping from site l to site l in the time interval t → t + 1, γ > 0 introduces a clockwise bias in the hopping probability, and (l,t) is a 061134-1

©2011 American Physical Society

ALEX DICKSON, S. M. ALI TABEI, AND AARON R. DINNER

PHYSICAL REVIEW E 84, 061134 (2011) (a)

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FIG. 1. (Color online) A schematic of the driven “oscillator.” A single particle (the white circle) hops on a periodic one-dimensional lattice with L sites. The hops are biased by both an intrinsic clockwise tendency (given by γ ) and a driving force that oscillates in direction with a period T .

0.6 0.5 0.4 0.3 0.2 0.1 0

where f controls the strength of the driving force, L is the number of sites in the loop, and T is the period of the driving force. We impose periodic boundary conditions on the lattice. Note that takes both positive and negative values and thus can promote either clockwise or counterclockwise entrainment depending on the values of γ and the s field as discussed further below. Straightforward trajectories with different values of f show different behaviors (Fig. 2). For values of the driving frequency f = 1/T much higher (lower) than the natural frequency (0 = 2γ /L), the observed oscillations are slower (faster) than f , and synchronization is not observed [Figs. 2(a) and 2(c)]. However, if f is close to 0 , the oscillations are frequency entrained to the driving force, and synchronization is observed [Fig. 2(b)]. Although entrainment can be observed in this way using individual trajectories, in the next section we describe how to exploit the discrete nature of the model to obtain infinite-time expectation values by diagonalizing biased transition matrices. III. OBTAINING LARGE DEVIATION FUNCTIONS

The large deviation function can be used to study general observables of the form [8] A(τ ) =

K−1

α(Ck ,Ck+1 ),

0

(2)

(3)

k=0

where Ck is the configuration of the system and K is the number of configuration changes undergone by the system in the time interval [0,τ ]. Here we choose Ck = {l,tc } to be the configuration of the system in space-time with l ∈ [0,L) denoting the position of the particle in space and tc ∈ [0,T )

2

4 Time (t/T)

(c)

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driving force that is periodic in time and space, 2π t 2π l cos , (l,t) = f cos L T

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Time (t/T)

FIG. 2. (Color online) Typical trajectories of the model. Five independent trajectories initiated from random starting configurations are shown. The trajectories are obtained with f = 0.1, γ = 0.25, L = 280, r = 0.01, and differing values of f as indicated. (a) Fast driving frequency (f = 20 ). The system cannot keep up with the driving force and exhibits less than one oscillation per driving cycle. (b) Close-to-natural oscillatory frequency (f = 40 /5). Although the driving frequency is slower than the natural frequency, we still observe frequency entrainment and synchronization. (c) Slow driving frequency (f = 20 /5). The system exhibits more than one oscillation per cycle.

denoting the time within the period of the driving force. Note that since both l and tc can have only a finite set of values, the number of possible configurations of the system is finite and equal to LT . Here, both l and tc are integer variables measured in units of lattice positions and time steps, respectively. This way, the system is guaranteed to change configuration exactly once in each time step even though the particle might remain

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ENTRAINMENT OF A DRIVEN OSCILLATOR AS A . . .


in the same position. A(τ ) is the net particle current if we use ⎧ ⎪ ⎨+1 for clockwise jumps, (4) α(Ck ,Ck+1 ) = −1 for counterclockwise jumps, ⎪ ⎩0 otherwise.

P (π,s) ∝ P (π,0) exp[−sA(τ )],

0.5 Ω(s)/Ωmax

Large deviation functions reveal information about values of this observable that are far away from their infinite-time expectation value. This is done by reweighting trajectories using a biasing pressure s:

1

where W (C → C) is the transition rate from C to C and

C =C W (C → C ) is the exit rate from state C. Note that for our purposes the exit rate is unity for all states C since the space-time configuration of the system is guaranteed to change every time step. Although numerical techniques exist to sample trajectories from these biased ensembles [12,14,15], here we can exploit the finite size of the system and construct the complete W (s) matrix for many values of s and diagonalize it to obtain the highest eigenvalue, the logarithm of which is the large deviation function ψ(s) [14]. Since we consider states in spacetime, the transition matrix has (LT )2 elements. We have found that the matrix diagonalization routine performs well for values of LT 1700 after which issues with numerical precision arise. Nevertheless, this range is sufficient to illustrate the interesting behavior of the model. IV. RESULTS A. Strong driving force

We first examine a set of parameters with a large driving amplitude, f = 0.4, γ = 0.05, and r = 0.01, and the size of the lattice is L = 28. The natural rotation frequency of the oscillator is given by 0 = 2γ /L, and we examine values of T in the range [30,120], corresponding to 20 < f < 90 . As mentioned above, we obtain the large deviation function ψ(s) directly from the diagonalization of W (s). We then numerically differentiate with respect to s to obtain ψ (s), which is equal to the opposite of the long-time expectation of the current within the biased ensemble: 1 ψ (s) = − lim At s . (7) t→∞ t The expectation value of the rotational frequency is then = −ψ (s)/L. Physically, negative (positive) s values bias the system to rotate in the clockwise (counterclockwise) direction, and s = 0 is the unbiased system. In Fig. 3, we show the expectation values of the current as a function of s. We plot our results normalized by max = 1/L, the maximum allowable frequency

Ωf = 1/30 Ωf = 1/34 Ωf = 1/38 Ωf = 1/44 Ωf = 1/53 Ωf = 1/65

-0.5

(5)

where π is a trajectory of time length τ and P (π,0) is the unbiased probability of the trajectory. This biased ensemble can be accessed using modified dynamics, where, for discrete systems, transitions are given by the s-dependent matrix W (s) with elements W (C → C)e−sα(C ,C) C = C , C|W (s)|C =

(6) C = C, C =C W (C → C )

0

-1 -4

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-2

-1

0 s

1

2

3

4

FIG. 3. (Color online) The measured rotational frequency (/ max ) for differing values of f with max = 1/L = 1/28. In each curve there is a plateau for s < 0 corresponding to clockwise entrainment and another for s > 0 corresponding to counterclockwise entrainment. The position of each plateau is at = ±f , and the widths of the plateaux approach infinity as f approaches max .

of rotation, corresponding to one clockwise jump per unit time. The curves in Fig. 3 should be interpreted as follows. Starting with the unbiased system at s = 0, each of these curves reveals the nonzero expectation of the current, shown by their intersections with the s = 0 axis. As we move toward larger negative values of s, the system makes a transition into an entrainment regime where is equal to f and does not change even as we increase the bias strength. Eventually, the biasing strength becomes large enough that this plateau ends, and clockwise motion is favored (eventually reaching the limit = max ). For each curve, there is also a plateau corresponding to counterclockwise entrainment (since it is the product sA that enters the weights of the trajectories of the modified dynamics so that the signs of s and A are coupled). We can compare the strength of the biasing field required to drive the system into clockwise and counterclockwise entrainment using the position of the peak in the derivative of the curves shown in Fig. 3. If γ = 0, then these two field strengths would be equal by symmetry, but since γ > 0, a larger biasing field is required for counterclockwise entrainment (s ∗ = 0.52 for = 1/30) than for clockwise entrainment (s ∗ = −0.40 for = 1/30). Because the two entrainment behaviors derive from the same physics, for simplicity we focus our analysis on clockwise entrainment. In Fig. 4 we plot a map of for different values of s and f . The color map shows the different phases of the system: > f , a faster rotation than the driving force, is shown in white; < f , a slower rotation than the driving force, is shown in black; and = f , entrainment to the driving force, is shown in red online (gray in print). As f approaches max , the entrainment regions become larger as it requires a larger clockwise bias to rotate faster than the driving force. For f = max , the entrainment region stretches to s = −∞ as it is physically impossible to rotate faster than the driving force. We now investigate the smoothness of the transitions in and out of the entrainment regions in Fig. 3 as a function of the finite

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ln(M) = 0.80 ln(T) -2.9

M = max(d2ψ/ds2)

10 8 6 4 -4

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0.2 10

FIG. 4. (Color online) A map of as a function of bias strength s and driving frequency is shown for a relatively strong driving force. The three dynamical phases of the system are shown: the entrainment region where = f is shown in red online (gray in print) and the region where the oscillator is going faster (slower) than the driving force is shown in white (black). In each curve of Fig. 3, the region to the left of the s < 0 plateau corresponds to the white region of this figure, the plateau to the red region, and the right of the plateau to the black region.

size of the system to determine if the derivatives tend toward singularities in analogy to numerical studies of equilibrium phase transitions. Here we take the limits (T ,L → ∞) while keeping the ratio L/T constant. This is equivalent to keeping constant the ratio of the driving frequency and the natural frequency (f / 0 ). This can be viewed as a form of continuum limit. Typically, when taking the L → ∞ limit, parameters that control the movements of the particle (such as γ and f ) should scale with L to ensure that the particle maintains the same oscillatory frequency (in cycles per time step). This is not possible in the model used here since the particle is only allowed to move at most one space per time step. Instead, we increase the number of time steps (T) along with L, keeping γ and f constant. In this way, although

T=12; L = 8 T=24; L=16 T=48; L=32

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ψ′′(s)

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FIG. 5. (Color online) (s)/ max for a set of curves with max / f = T /L = 3/2. Inset: d 2 ψ/ds 2 = −(1/ max )(d/ds) in the region of the entrainment transition.

20

50 T

FIG. 6. The height of the peak of d 2 ψ/ds 2 (shown in the inset of Fig. 5) as a function of T . The fit of this data to the function ln(M) = A1 ln(T ) + A2 has a χ 2 value of 1.1 × 10−5 with A1 = 0.800 and A2 = −2.86.

is changing, we can use a normalized frequency (/ max ), which is conserved as L and T increase. In Fig. 5 we plot a set of curves ranging from T = 12 (least sharp) to T = 48 (sharpest), all with L = 2T /3. In the inset, we plot −(1/ max )[d(s)/ds], which is equal to the second derivative of the large deviation function. It is essentially a susceptibility. We see in the figure that the transitions in and out of entrainment get sharper as the system size grows. In Fig. 6 we plot the value of the peak of the second derivative (M) as a function of T . This curve can be fit to the form ln(M) = A1 ln(T ) + A2 with a positive value of A1 , demonstrating that as T → ∞, the derivative of the expectation value of the current will also become infinite. In other words, there is evidence of a singularity. B. Weak driving force and an analytical investigation of the plateau

In the previous section, we diagonalized the transfer matrix numerically and found a plateau in the dynamical observable (the expectation value of the rotational frequency) as a function of the biasing pressure s. We now use the existing analogy between the transition matrix formalism and the Hamiltonian operator in quantum mechanics to obtain a better understanding of the observed plateau. Specifically, we note that similar plateaux are observed in a number of quantum magnetic systems in which the magnetization plays the role of the current and an external magnetic field plays the role of the biasing pressure [16,17]. Understanding the mechanism of these plateaux in frustrated quantum magnetic systems has recently attracted substantial theoretical attention. In quantum mechanics at zero temperature, the system populates the ground state, corresponding to the smallest eigenvalue of the Hamiltonian operator. If there exists a tunable parameter (such as the magnetic field) that changes the energy eigenvalues such that quantum fluctuations become larger than the distance between the energy levels, then the system undergoes a phase transition [18]. Similarly, the behavior of a dynamical system at long times is described by the largest eigenvalue of the

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FIG. 7. (Color online) A map of as a function of bias strength s and f is shown for a weak driving force. The three dynamical phases of the system are shown: the entrainment region where = f is shown in red online (gray in print) and the region where the oscillator is going faster (slower) than the driving force is shown in white (black). The s = 0 axis and f = 0 are shown as dotted lines to show that the entrainment region intersects the s = 0 axis when the driving frequency equals the natural frequency of the oscillator.

where

-2.5

-2

-1.5 s

-1

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0

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1

=

2π l 1 r 2π tc cos e∓s − ± γ + f cos 2 2 T L (9)

and |l ,tc is the basis vector with the element corresponding to l = l and tc = tc equal to 1 and the others equal to 0. We can then further break the operator W (s) into two parts: an operator independent of f and an operator linear in f , W (s) = W0 (s) + f W1 (s).

(10)

Following the Bethe method [19], we construct a state |k,ω from |l,tc of the form L,T

|k,ω =

eikl eiωtc |l,tc

(11)

l=1,tc =1

such that in the absence of the external driving force (f = 0), W0 (s) is diagonalized in the space of |k,ω . If we apply W0 (s) to |k,ω , we have W0 |k,ω = e−iω [r + e−ik σ0+ (s) + eik σ0− (s)]|k,ω , where σ0± (s) =

1 r − ±γ 2 2

e∓s

(12)

(13)

and the eigenvalues of W0 are given by 0 (s) = e−iω [r + e−ik σ0+ (s) + eik σ0− (s)]. Ek,ω

(14)

Now, if we make f nonzero and apply W (s) to |k,ω , we get

+ (s)|l + 1,tc + 1 W (s)|l,tc = r|l,tc + 1 + σl,t c − (s) |l − 1,tc + 1 , +σl,t c

-3

FIG. 8. (Color online) (s) for a set of curves with a weak driving force. These are obtained using L = 28 and varying values of f . These are similar to the curves from the strong driving force case (Fig. 3) except the plateaux are much thinner.

± (s) σl,t c

dynamical transition operator. The eigenvalue spectrum of the dynamical system can be either varied by changing the biasing field s or the system parameters. In this way, characterizing the stationary and nonstationary states and the coupling between them allows us to understand the dynamical transitions in our stochastic dynamical system. In the absence of the driving force f , we are able to obtain the eigenvalues and eigenstates of the transition operator analytically. Therefore, if we consider a relatively weak driving force, we can apply perturbation theory to show that f creates a coupling between the eigenstates of the unforced oscillator. To this end, we study a weaker driving force with the parameters γ = 0.25, f = 0.1, and r = 0.01. These choices enable comparisons to perturbation theory results that use the f = 0 solution as a ground state. We can also now study the f ≈ 0 regime since the natural rotation frequency of the oscillator is much faster: 0 = 2γ /L = 1/56, resulting in a natural rotation period of 56 time steps. Biased transition matrices for driving periods in this neighborhood are now small enough to diagonalize. In Fig. 7 we show the phase diagram for this set of parameters. The entrainment region is much thinner as expected for smaller driving amplitudes. At f = 0 , the entrainment region intersects the s = 0 axis. Note that a cross section of this plot along the s = 0 axis would reveal a “detuning” plot that shows the transition, as f increases, from the > f phase, through the “entrained” phase, and into the < f phase. curves for these parameters are plotted in Fig. 8 for different values of f . The evolution of the system under the biased probability transition matrix defined in Eq. (6) can be described as follows:

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(8) 061134-5

W (s)|k,ω = e−iω [r + e−ik σ0+ (s) + eik σ0− (s)]|k,ω 1 + f e−iω e−s−ik e−2πi(ζ /T +ξ/L) 4 ζ,ξ ∈{−1,1} × |k+2πξ/L,ω+2πζ /T


1 − f e−iω es+ik e−2πi(ζ /T −ξ/L) 4 ζ,ξ ∈{−1,1} × |k+2πξ/L,ω+2πζ /T ,

(15)

where the sums are over the four possible combinations of values of ζ and ξ : (−1, − 1), (−1,1), (1, − 1), and (1,1). Equation (15) shows that the external driving force f introduces a coupling between the states specified by

f

2

(2) Ek,ω (s)

=f

2


the quantum numbers k and ω and their nearest neighbors k ± 2π/L and ω ± 2π/T . In the case in which f is weak, we use standard nondegenerate perturbation theory [20] to obtain the eigenvalues Ek,ω (s) of W (s) to the second order in f . They take the form (2) 0 (s) + f 2 Ek,ω (s), Ek,ω (s) ≈ Ek,ω

where

k,ω |W1 (s)k+ 2ξ π ,ω+ 2ζ π k+ 2ξ π ,ω+ 2ζ π W1 (s)|k,ω

ζ,ξ ∈{−1,1}

0 0 Ek,ω (s) − Ek+ (s) 2ξ π ,ω+ 2ζ π

L

T

L

L

where the sum is over the same (ζ,ξ ) pairs as the sums in Eq. (15). Note that W (s) is not a Hermitian operator. By definition, the expectation value of the observed frequency is given by d[log Emax (s)] = −A = . max ds

(18)

(16)

T

(17)

,

T

other. To be more precise, the second-order correction fails when the magnitude of the off-diagonal transition matrix elements k,ω |W1 (s)|k±2π/L,ω±2π/T between the neighboring states become larger than the magnitude of the eigenvalue difference of the corresponding states [20]. In Fig. 10, we show the interval of s in which |W (s)| k±2π/L,ω±2π/T k,ω 1 f 0 1 0 Ek,ω (s) − Ek±2π/L,ω±2π/T (s)

Emax is the largest eigenvalue of the transition operator W (s), which specifies the large deviation function as discussed in Sec. III. Our choice of notation is such as to emphasize the analogy to energy in zero-temperature quantum systems. The result is shown in Fig. 9. As we can see, outside of the entrainment regime, the second-order perturbation theory agrees well with the exact diagonalization results, but this does not hold inside the entrainment regime. Looking at the second-order contribution of the eigenvalues calculated in Eq. (17), we see that it is not well-defined if the eigenvalue levels in the denominator get very close to each

(19)

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0 −2

Exact Diagonalization

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Perturbation Theory (PT) Close-to-Degenerate PT

−1.5

−1

−0.5

0

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s FIG. 9. (Color online) / max curves obtained by exact diagonalization, conventional second-order perturbation theory, and closeto-degenerate perturbation theory including the transition matrix of all the neighboring states. These are obtained using L = 28 and T = 34 and the other parameters used for the weak driving force case.

FIG. 10. (Color online) The vertical axis is a measure of the relative strength of coupling between different states with respect to the (energy) difference between the states. This is defined by dividing k,ω |W1 (s)|k±2π/L,ω±2π/T by the eigenvalue difference 0 0 (s) − Ek±2π/L,ω±2π/T (s). We have plotted for all the E 0 = Ek,ω possible values of k and ω. When values exceed one, we expect strong coupling between the ground state and the excited states, and a close-to-degenerate perturbation theory is needed to accurately determine the energy levels. This corresponds to the entrainment region. The results shown are obtained using L = 28 and T = 34.

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for an amplitude of f = 0.1. This shows that in the entrainment region the distance between the levels becomes smaller than the coupling between the states. To reproduce the exact diagonalization results in the entrainment region, we used a rigorous extension of close-todegenerate perturbation theory [20]. Strong coupling between any two states can cause the perturbation results to diverge. In the standard approach [20], one state is assumed to be close to degenerate; here, we instead allow for the whole spectrum of the transition matrix eigenstates to be close to degenerate. This is equivalent to diagonalizing a matrix in which the 0 diagonal terms are Ek,ω (s) and the nonzero off-diagonal terms are k,ω |W1 (s)|k±2π/L,ω±2π/T . In Fig. 9, it is shown that when the whole spectrum is incorporated to second order in f , the results agree with the exact diagonalization results, and the plateau region is correctly reproduced. This result suggests that we can use the theory as the basis for a physical interpretation of stochastic entrainment within the large deviation framework. In particular, using an eigenanalysis helps us to build an analogy with phase transitions in quantum systems [13,18,21]. In this vein, we see that in an s-biased dynamical system, the largest eigenvalue of W (s) is associated with the ground state, and the other eigenvalues are associated with excited states. Equation (15) shows that the external driving force f introduces coupling between these states. The quantum analogy can be understood as follows: by increasing f , the magnitude of the off-diagonal couplings induced by f can become larger than the differences between the energy levels, enabling intrinsic quantum fluctuations among the eigenstates. The biasing field s then acts as a tuning parameter that can both enhance the coupling between states and bring eigenvalues closer together. Therefore, by varying s we can enter a regime in which the intrinsic fluctuations become larger than the differences between the (energy) levels, and the system transits among states. It is in this region in which the oscillator exhibits the frequency of the driving force. This behavior is not unique to the presence of an s-biased field, and it can be seen for certain values of T where the entrainment region intersects the s = 0 axis (see Fig. 7). The advantage of the s-biased approach is that if, for a given value of T , the system does not entrain, by biasing the s field we can push the system into the entrainment region by pushing the states to close-to-degenerate states. This can provide insights into the stochastic dynamics (e.g., transient entrainment to the driving force) even when there is no transition on the s = 0 axis.

V. DISCUSSION

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Here, we examined the entrainment of an oscillator to a periodic driving force using the large deviation function formalism. We considered a simple model of a single particle hopping on a ring that is coupled to a periodic external driving force through modification of the left and right hopping probabilities. By studying the response of the system to a biasing field, we revealed dynamical phase transitions that cause singularities in the derivative of the expectation value with respect to the strength of the biasing field. We further showed that these transitions were manifestations of entrainment of the oscillator to the external driving force. Conceptually, this is a significant result since we are now able to incorporate entrainment transitions into a general statistical mechanical framework. As such, we have a language with which to compare different entrainment transitions to each other and to properties of other dynamical phase transitions, such as the motion to nonmotion transitions in the asymmetric exclusion process and glassy systems. Here, due to the discreteness and the small size of the system, we were able to calculate large deviation functions directly from the biased transition matrix. For more complicated systems, such as those with continuous degrees of freedom, this approach will not be feasible. However, there exist techniques to calculate the large deviation function by sampling a biased ensemble of trajectories that are applicable to more complicated systems [12,14,15]. It will be interesting to see whether the shape of the phase diagram obtained here is general for a broad class of single oscillators coupled to an external driving force. In this work we focused on a single biasing field that is coupled to the rotational frequency of the oscillator. Under the influence of this field at a given s value, the oscillator can only be made to oscillate faster or slower. Alternatively, one could employ a biasing field that is coupled to the difference in phase of a set of coupled oscillators. This would allow for the description of emergent synchronization into the statistical mechanical framework used here.

ACKNOWLEDGMENTS

We thank Martin Tchernookov and M. J. P. Gingras for useful discussions. S.M.A.T. was supported by the Human Frontier Science Program. A.D. was supported by a fellowship from the Natural Sciences and Engineering Research Council. This work was also supported by a grant from the W. M. Keck Foundation.

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