Bounding energy growth in frictionless stochastic oscillators

This paper presents analytical and numerical results on the energetics of nonharmonic, undamped, single-well, stochastic oscillators driven by additive Gaussian white noises. The absence of damping and the action of noise are responsible for the lack of stationary states in such systems. We explore the properties of average kinetic, potential, and total energies along with the generalized equipartition relations. It is demonstrated that in frictionless dynamics, nonequilibrium stationary states can be produced by stochastic resetting. For an appropriate resetting protocol, the average energies become bounded. If the resetting protocol is not characterized by a finite variance of renewal intervals, stochastic resetting can only slow down the growth of the average energies but it does not bound them. Under special conditions regarding the frequency of resets, the ratios of the average energies follow the generalized equipartition relations.

Figure 1

Time dependence of average energies for the frictionless, uninterrupted motion in the parabolic potential ($n=1$) with $(h_x,h_v)=(0,1)$ [top panel, (a)], $(h_x,h_v)=(1,0)$ [middle panel, (b)], and $(h_x,h_v)=(1,1)$ [bottom panel, (c)]. Other parameters: $k=1$ and $m=1$. Solid lines present theoretical formulas [see Eqs. (17, 18, 19)] while the dot-dashed line presents $\langle \mathcal{E}\left(t\right)\rangle /2$ [see Eqs. (21) and (22)].

Figure 2

The same as in Fig. 1 for $k=4$ and $m=1$. Solid lines present theoretical formulas; see Eqs. (17, 18, 19). The dot-dashed line presents $\langle \mathcal{E}\left(t\right)\rangle /2$; see Eqs. (21) and (22).

Figure 3

Time dependence of average energies for the frictionless, uninterrupted motion in the quartic potential ($n=2$) with $(h_x,h_v)=(0,1)$ [top panel, (a)], $(h_x,h_v)=(1,0)$ [middle panel, (b)], and $(h_x,h_v)=(1,1)$ [bottom panel, (c)]. Other parameters: $k=1$ and $m=1$. The solid line in the top panel depicts the theoretical linear scaling of the average total energy $\langle \mathcal{E}\left(t\right)\rangle$; see Eq. (19). Dashed and dot-dashed lines present $\frac{2}{3}\langle \mathcal{E}\left(t\right)\rangle$ and $\frac{1}{3}\langle \mathcal{E}\left(t\right)\rangle$ scalings; see Eqs. (21) and (22).

Figure 4

Time dependence of average energies for the parabolic [top panel, (a)] and quartic [bottom panel, (b)] potentials with $(h_x,h_v)=(0,1)$. The reset is performed every $\tau =4$, i.e., $f\left(\tau \right)=\delta (\tau -4)$. Other parameters: $k=1$ and $m=1$. Solid, dashed, and dot-dashed lines present ${\langle \mathcal{E}\rangle }_{\max }$ with $\frac{1}{2}{\langle \mathcal{E}\rangle }_{\max }$, $\frac{1}{2}{\langle \mathcal{E}\rangle }_{\max }$ [top panel (a)], and ${\langle \mathcal{E}\rangle }_{\max }$ with $\frac{2}{3}{\langle \mathcal{E}\rangle }_{\max }$, $\frac{1}{3}{\langle \mathcal{E}\rangle }_{\max }$ [bottom panel (b)]; see Eqs. (21) and (22).

Figure 5

Ratio of average energies $\langle {\mathcal{E}}_k\rangle /\langle \mathcal{E}\rangle$ and $\langle {\mathcal{E}}_p\rangle /\langle \mathcal{E}\rangle$ at $t=\mathrm{\Delta }$ for the parabolic [top panel, (a)] and quartic [bottom panel, (b)] potentials with $(h_x,h_v)=(0,1)$. The reset is performed every $\mathrm{\Delta }$, i.e., $f\left(\tau \right)=\delta (\tau -\mathrm{\Delta })$. Other parameters: $k=1$ and $m=1$. Solid, dashed, and dot-dashed lines present theoretical ratios given by Eqs. (21) and (22), i.e., $1/2$ [top panel, (a)] and $2/3$, $1/3$ [bottom panel, (b)].

Figure 6

Time dependence of average energies for the parabolic ($n=1$) potential with $(h_x,h_v)=(0,1)$ [top panel, (a)], $(h_x,h_v)=(1,0)$ [middle panel, (b)], and $(h_x,h_v)=(1,1)$ [bottom panel, (c)]. Renewal intervals $\tau$ are distributed according to the half-normal distribution, see Eq. (31). Different curves correspond to various types of energies $\langle \mathcal{E}\left(t\right)\rangle ,\langle {\mathcal{E}}_p\left(t\right)\rangle ,\langle {\mathcal{E}}_k\left(t\right)\rangle$. Solid lines depict the asymptotic energy $\langle \mathcal{E}\rangle$ given by Eq. (32), while the dash-dotted line in the bottom panel (c) shows $\frac{1}{2}\langle \mathcal{E}\rangle$. Other parameters: $m=1$ and $k=1$.

Figure 7

The same as in Fig. 6 for the quartic ($n=2$) potential. The solid line in the top panel (a) depicts the asymptotic energy $\langle \mathcal{E}\rangle$ given by Eq. (32).

Figure 8

Nonequilibrium stationary densities $P(x,v)$ for the parabolic ($n=1$) potential with $(h_x,h_v)=(0,1)$ [top left panel, (a)], $(h_x,h_v)=(1,0)$ [top right panel, (b)], and $(h_x,h_v)=(1,1)$ [bottom panel, (c)]. Renewal intervals $\tau$ are distributed according to the half-normal distribution, see Eq. (31). Other parameters: $m=1$ and $k=1$.

Figure 9

Marginal nonequilibrium stationary densities corresponding to Fig. 8, i.e., $P\left(x\right)$ [top panel, (a)] and $P\left(v\right)$ [bottom panel, (b)] for the parabolic ($n=1$) potential. Different curves correspond to different values of $h_x$ and $h_v$. Solid lines represent $a_{x}exp(-b_x{x}^2)$ and $a_{v}exp(-b_v{v}^2)$ fits. Renewal intervals $\tau$ are distributed according to the half-normal distribution, see Eq. (31). Other parameters: $m=1$ and $k=1$.

Figure 10

Ratio of average energies for the parabolic ($n=1$) potential with $(h_x,h_v)=(0,1)$. Different curves correspond to $\langle {\mathcal{E}}_p\rangle /\langle \mathcal{E}\rangle$ and $\langle {\mathcal{E}}_k\rangle /\langle \mathcal{E}\rangle$. Dashed and dot-dashed lines depict theoretical ratios predicted by Eqs. (21) and (22). Renewal intervals $\tau$ are distributed according to the half-normal distribution, see Eq. (31). Other parameters: $m=1$ and $k=1$.

Figure 11

The same as in Fig. 10 for the quartic ($n=2$) potential with $(h_x,h_v)=(0,1)$ [top panel, (a)], $(h_x,h_v)=(1,0)$ [middle panel, (b)], and $(h_x,h_v)=(1,1)$ [bottom panel, (c)]. Renewal intervals $\tau$ are distributed according to the half-normal distribution; see Eq. (31).

Figure 12

Time dependence of average energies for the parabolic potential with $(h_x,h_v)=(0,1)$ and the Pareto distribution, see Eq. (38), of renewal intervals with $1<\alpha <2$ [top panel, (a)] and $0<\alpha <1$ [bottom panel, (b)]. Various curves correspond to various values of the exponent $\alpha$. Other parameters: $\delta =0.01$, $k=4$, and $m=1$. Solid lines present the theoretical $t^{2-\alpha }$ scaling.