Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case
A novel experimental paradigm and a novel modelling approach are presented to investigate oscillatory human motor performance by means of a key concept from condensed matter physics, namely, thermodynamic state variables. To this end, in the novel experimental paradigm participants performed pendu...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case / J.M. Gordon, S. Kim, T.D. Frank // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 34001: 1–6. — Бібліогр.: 16 назв. — англ. |
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irk-123456789-1562232019-06-19T01:25:32Z Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case Gordon, J.M. Kim, S. Frank, T.D. A novel experimental paradigm and a novel modelling approach are presented to investigate oscillatory human motor performance by means of a key concept from condensed matter physics, namely, thermodynamic state variables. To this end, in the novel experimental paradigm participants performed pendulum swinging movements at self-selected oscillation frequencies in contrast to earlier studies in which pacing signals were used. Moreover, in the novel modelling approach, a canonical-dissipative limit cycle oscillator model was used with a conservative part that accounts for nonharmonic oscillator components in contrast to earlier studies in which only harmonic components were considered. Consistent with the Landau theory of magnetic phase transitions, we found that the oscillator model free energy decayed when participants performed oscillations further and further away from the Hopf bifurcation point of the canonical-dissipative limit cycle oscillator Представлено нову експериментальну парадигму та новий пiдхiд до моделювання роботи осциляторного людського двигуна з допомогою ключового концепту фiзики конденсованої речовини, а саме — змiнних термодинамiчного стану. Для цього в новiй експериментальнiй парадигмi учасники здiйснювали маятниковi коливнi рухи з особисто вибраними частотами, що вiдмiнне вiд попереднiх дослiджень, де використовувалися синхронiзуючi сигнали. У новому пiдходi моделювання використано канонiчно-дисипативну границю циклiчно осциляцiйної моделi з консервативною частиною, що враховує негармонiчнi коливнi компоненти — у цьому iнша вiдмiннiсть вiд попереднiх дослiджень, у яких розглядалися лише гармонiчнi компоненти. В узгодженнi з теорiєю Ландау магнiтних фазових переходiв ми знайшли, що вiльна енергiя осциляцiйної моделi зменшується, коли учасники пiдтримують коливання з вiддаленням вiд точки бiфуркацiї Хопфа для канонiчно-дисипативної границi циклiчного осцилятора. 2016 Article Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case / J.M. Gordon, S. Kim, T.D. Frank // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 34001: 1–6. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 05.40.Jc, 05.45.-a, 05.70.Ce, 05.70.Ln, 64.70.qd, 87.19.rs DOI:10.5488/CMP.19.34001 arXiv:1611.00285 http://dspace.nbuv.gov.ua/handle/123456789/156223 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
A novel experimental paradigm and a novel modelling approach are presented to investigate oscillatory human
motor performance by means of a key concept from condensed matter physics, namely, thermodynamic state
variables. To this end, in the novel experimental paradigm participants performed pendulum swinging movements at self-selected oscillation frequencies in contrast to earlier studies in which pacing signals were used.
Moreover, in the novel modelling approach, a canonical-dissipative limit cycle oscillator model was used with a
conservative part that accounts for nonharmonic oscillator components in contrast to earlier studies in which
only harmonic components were considered. Consistent with the Landau theory of magnetic phase transitions,
we found that the oscillator model free energy decayed when participants performed oscillations further and
further away from the Hopf bifurcation point of the canonical-dissipative limit cycle oscillator |
| format |
Article |
| author |
Gordon, J.M. Kim, S. Frank, T.D. |
| spellingShingle |
Gordon, J.M. Kim, S. Frank, T.D. Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case Condensed Matter Physics |
| author_facet |
Gordon, J.M. Kim, S. Frank, T.D. |
| author_sort |
Gordon, J.M. |
| title |
Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case |
| title_short |
Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case |
| title_full |
Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case |
| title_fullStr |
Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case |
| title_full_unstemmed |
Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case |
| title_sort |
linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative fokker-planck equation approach involving potentials beyond the harmonic oscillator case |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/156223 |
| citation_txt |
Linear non-equilibrium thermodynamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case / J.M. Gordon, S. Kim, T.D. Frank // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 34001: 1–6. — Бібліогр.: 16 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT gordonjm linearnonequilibriumthermodynamicsofhumanvoluntarybehavioracanonicaldissipativefokkerplanckequationapproachinvolvingpotentialsbeyondtheharmonicoscillatorcase AT kims linearnonequilibriumthermodynamicsofhumanvoluntarybehavioracanonicaldissipativefokkerplanckequationapproachinvolvingpotentialsbeyondtheharmonicoscillatorcase AT franktd linearnonequilibriumthermodynamicsofhumanvoluntarybehavioracanonicaldissipativefokkerplanckequationapproachinvolvingpotentialsbeyondtheharmonicoscillatorcase |
| first_indexed |
2025-07-14T08:40:55Z |
| last_indexed |
2025-07-14T08:40:55Z |
| _version_ |
1837611044882612224 |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 3, 34001: 1–6
DOI: 10.5488/CMP.18.34001
http://www.icmp.lviv.ua/journal
Rapid communication
Linear non-equilibrium thermodynamics of human
voluntary behavior: a canonical-dissipative
Fokker-Planck equation approach involving
potentials beyond the harmonic oscillator case
J.M. Gordon1, S. Kim1, T.D. Frank1,2
1 CESPA, Department of Psychology, University of Connecticut, 406 Babbidge Road, Storrs, CT 06269, USA
2 Department of Physics, University of Connecticut, 2152 Hillside Road, Storrs, CT 06269, USA
Received June 7, 2016
A novel experimental paradigm and a novel modelling approach are presented to investigate oscillatory human
motor performance by means of a key concept from condensed matter physics, namely, thermodynamic state
variables. To this end, in the novel experimental paradigm participants performed pendulum swinging move-
ments at self-selected oscillation frequencies in contrast to earlier studies in which pacing signals were used.
Moreover, in the novel modelling approach, a canonical-dissipative limit cycle oscillator model was used with a
conservative part that accounts for nonharmonic oscillator components in contrast to earlier studies in which
only harmonic components were considered. Consistent with the Landau theory of magnetic phase transitions,
we found that the oscillator model free energy decayed when participants performed oscillations further and
further away from the Hopf bifurcation point of the canonical-dissipative limit cycle oscillator.
Key words: physics of life, human behavior, canonical-dissipative systems, thermodynamic state variables
PACS: 05.40.Jc, 05.45.-a, 05.70.Ce, 05.70.Ln, 64.70.qd, 87.19.rs
In view of the success of physics in general and condensed matter physics in particular, the ques-
tion has been debated whether concepts from physics and condensed matter physics can be generalized
to describe not only equilibrium systems but also non-equilibrium systems (e.g. [1, 2]). The canonical-
dissipative (CD) approach to non-equilibrium systems [3–6] in this context plays an important role be-
cause it exhibits key features of condensed matter physics such as Hamiltonian mechanics and thermo-
dynamic state variables [7]. Moreover, in a series of recent experimental studies it has been shown that
the CD approach can be applied to characterize the performance of humans — as paradigmatic non-
equilibrium systems — in simple oscillatory motor control tasks [8–10]. However, these earlier studies
were subjected to two severe limitations. First, the experimental studies involved a pacing signal which
is inconsistent with the model of an autonomous limit-cycle oscillator. Although in the study by Kim et
al. [10], the impact of the pacing signal was greatly reduced, the question arises whether or not the ex-
perimental design can be improved to do without any pacing signal. Second, all three studies used the
standard CD oscillator model exhibiting the conservative part of a harmonic oscillator. In this regard, the
question arises how to go beyond the harmonic oscillator case. In what follows we will report an experi-
mental design inwhich the pacing signal was completely absent. Moreover, wewill consider conservative
parts of the CD modelling approach that correspond to nonharmonic oscillators.
In the experiment participants were sitting in a chair with their right arm fixed to a custom armrest,
allowing them to swing a pendulum (40 cm rod, cylindrical weight positioned 2.5 cm from end, 303 g total
weight) with resonant frequency of 1 Hz (6.28 rad/s) with their right hand about the wrist. Goniometers
were attached to the arm to capture the wrist angle (measured in radiants) while swinging the pendulum
(Biometrics, 100 Hz sampling rate). For the duration of each one-minute trial, participants were instructed
© J.M. Gordon, S. Kim, T.D. Frank, 2016 34001-1
http://dx.doi.org/10.5488/CMP.18.34001
http://www.icmp.lviv.ua/journal
J.M. Gordon, S. Kim, T.D. Frank
to swing at one of three paces: comfortable, fast, or slow (3 conditions, 3 repetitions each, presentation
randomized). Rather than synchronize their swinging to an external signal (i.e., a metronome) for refer-
ence, participants determined their own pacing for all trials, scaled relative to their natural (“comfort-
able”) swinging frequency. Practice trials ensured that participants could generate 3 different swinging
paces and sustain them for one minute each. Practice began with the “comfortable” condition. A tablet
(Android, Nexus 10) was used to approximate the participant pacing in beats per minute (BPMs), where a
“beat”was marked at the furthest point of the pendulum’s arc. Average BPMwas also tracked for fast and
slow conditions. “Fast” swinging was required to be at least 20% faster than their comfortable pace, and
“slow”was likewise required to be at least 20% slower in speed. If pacing during a slow or fast trial failed
to exceed this 20% difference, participants were notified and prompted to increase or decrease their pace
accordingly. After practice, most participants completed the experiment successfully and without further
corrections from the experimenter. The experimental design allowed us to investigate and manipulate
voluntary, oscillatory human motor performance without the use of any external pacing signal. To this
end, the oscillatory goniometer signal was evaluated.
Let x denote goniometer (wrist angle) signal in radiants and let v denote the corresponding velocity
v = dx/dt . Let r = (x, v) denote the state vector of the oscillatory system and H(x, v) a Hamiltonian
function that will be specified later. The Langevin equation [11] of the CD model for the self-oscillator
reads [3, 7, 12] d
dt
r = I−γ∇∇∇vΦ+
(
0p
DΓ(t )
)
, (1)
where I = (∂H/∂v, −∂H/∂x) (from a dynamical system’s perspective) is the conservative “force” acting on
the oscillator. In equation (1)∇∇∇v = (0,∂/∂v) is the incomplete Nabla operator and Φ=Φ(H) is a function
of H . The parameter D Ê 0 denotes the diffusion coefficient and is composed of the damping parameter
γÊ 0 and the noise amplitude θ Ê 0 likeD = γθ. Finally, Γ(t ) denotes a Langevin force [11] normalized like〈
Γ(t )Γ(t ′)
〉 = 2δ(t − t ′). Here and in what follows δ(z) represents the Dirac delta function and 〈·〉 stands
for ensemble averaging. In the conservative case (i.e., for γ = 0 ⇒ D = 0) equation (1) corresponds to a
Hamiltonian dynamics. Therefore, we interpret H as Hamiltonian function of the oscillator. The proba-
bility density of the state variables x and v is defined by P (x, v, t ) = 〈
δ
(
x −x(t )
)
δ
(
v − v(t )
)〉. From equa-
tion (1) it follows that P (x, v, t ) satisfies the so-called free energy Fokker-Planck equation [7] of the form
∂
∂t
P =−∇∇∇· IP +∇∇∇·M ·P∇∇∇δFNE
δP
. (2)
In equation (2) we have the complete Nabla operator ∇∇∇ = (∂/∂x,∂/∂v) and δFNE/δP denotes the vari-
ational derivative of the functional FNE with respect to P . The functional FNE itself is one of the three(non-equilibrium) thermodynamic variables of the CD model and denotes the non-equilibrium free en-
ergy. The two other variables are the non-equilibrium internal energyUNL and the statistical entropy S
defined by [7]
S =−
∫
P lnP dxdv, UNL = 〈Φ〉 , FNE =UNL −θS. (3)
Moreover, M denotes a 2× 2 mobility matrix with coefficients M22 = γ and M j k = 0 otherwise. From
equation (3) it follows that θ plays the role of a non-equilibrium temperature. The stationary solution of
equation (2) in phase space (x, v) reads
P (x, v) = 1
Zxv
exp
[
−Φ
(
H(x, v)
)
θ
]
, (4)
where Zxv is a normalization constant. Let us introduce the probability density of the Hamiltonian func-tion H like P (H) = ∫
δ
(
H −H(x, v)
)
P (x, v)dxdv . We obtain
P (H) = 1
ZH
exp
[
−Φ(H)
θ
]
, (5)
where ZH is a normalization constant again. Note that the normalization constants Zxv and ZH are re-lated to each other [6, 9]. Importantly, P (H) is maximal at the energy value H for which Φ is minimal.
34001-2
Thermodynamics of voluntary behavior
Therefore,Φ(H) can be regarded as non-equilibrium potential of H . In addition, if we putΦ= (H−B)2/2,
where B is a parameter that will be discussed below, then with respect to P (H) the variables FNL , UNLand S defined by equation (3) can be calculated from B and θ like [9, 10]
UNE = θ
2
{
1− B exp[−B 2/(2θ)]
ZH
}
, S = ln ZH + UNE
H
θ
, FNE =−θ ln ZH (6)
with ZH = p
2πθw(B/
p
θ), where w(s) is the error function w(s) = ∫ s
−∞ exp(−z2/2)dz.
Let us dwell on the interpretation of Φ. Let us put γ> 0 but θ = 0 ⇒ D = 0 such that
d
dt
x = ∂H
∂v
,
d
dt
v =−∂H
∂x
−γ∂Φ
∂v
⇒ d
dt
Φ=−γ
(
∂Φ
∂v
)2
É 0. (7)
That is, Φ is a nonincreasing function of time. Frequently, the special case of a quadratic function
Φ(H) = (H −B)2
2
(8)
has been discussed in the literature (e.g. [3]). In this case, equation (7) reads
d
dt
x = ∂H
∂v
,
d
dt
v =−∂H
∂x
−γ∂H
∂v
(H −B) ⇒ d
dt
H =−γ
(
∂H
∂v
)2
(H −B). (9)
Let us assume H is bounded from below. For the sake of simplicity we assume that (i) H Ê 0 holds for
all x, v , (ii) H is a smooth function defined on the phase space x, v , and there exists at least one pair
x(min), v(min) such that H = 0. Furthermore, let us assume that if H > 0 holds then trajectories x(t ), v(t )
of equation (9) are such that ∂H/∂v = 0 holds only for a discrete set K of time points t1 < t2 < t3 . . . . Conse-
quently, provided that H > 0 and H ,B holds, then we have dH/dt < 0 if t is not in the set K . This implies
thatH andΦ exhibit asymptotically stablefixed points atH = 0 andΦ= B 2/2 forB É 0 andH = B andΦ=
0 for B > 0. In particular, for both cases, the potentialΦ converges to its minimum valueΦmin = minH (Φ).
So far, in applications of the model (1) to experimental research on human motor behavior, the focus
has been on the harmonic oscillator case [8–10]
H = v2
2
+ ω2x2
2
. (10)
Taking B as a control parameter of the deterministic oscillatory system (1) with θ = 0, then there is a Hopf
bifurcation point at B = 0. In the experiment, the parameter B reflects the intention of a participant to
swing (B > 0) or not to swing (B É 0) the pendulum. As far as the CD oscillator model is concerned, for
B É 0 equation (9) exhibits an asymptotically stable fixed point at (x, v) = (0,0) with H = 0. By contrast,
for B > 0 the fixed point (x, v) = (0,0) is an unstable focus. There exists a stable limit cycle characterized
by the harmonic oscillator equation d2x/dt 2 =−ω2x and an oscillation amplitude A given by ω2 A2/2 =
B [8]. Consequently, on the limit cycle we have x(t ) =ω−1
p
2B cos(ωt+ϕ), whereϕ is an arbitrary phase.
Let us generalize the considerations in order to consider nonharmonic limit cycles. To this end, we
consider Hamiltonian functions of the form
H = v2
2
+V (x), V (x) =V0(x)+V1(x). (11)
V0 is considered as baseline oscillator potential and V1 is considered as a correction term. However, wedo not require that V1 should denote a small correction term. By contrast, we require that the baselinepotential should be quadratic like
V0(x) = k1x +k2
x2
2
(12)
with k2 > 0, whereas V1 accounts for contributions different from the linear and quadratic cases. Inorder to make sure that V (x) is globally stable such that non-equilibrium thermodynamic variables FNE
34001-3
J.M. Gordon, S. Kim, T.D. Frank
andUNL and the statistical entropy S exist we require that V1(|x| →∞) = 0 should hold. In what follows,
we will use
V1(x) = exp(−x2/2)
N∑
j=2
k j+1x j+1. (13)
In fact, for our purposes any other functionV1 could be used provided thatV1(|x|→∞) = 0 holds andV1 islinearly independent with respect to the linear and quadratic functions x and x2, respectively. Let us con-
sider the conservative case γ= 0 ⇒ D = 0 again. From dr/dt = I and equations (11–13) it then follows that
d2
dt 2 x =−k1 −k2x −exp(−x2/2)
N∑
j=2
kn+1
[
(n +1)xn −xn+2] . (14)
Therefore, in the dissipative, deterministic case γ> 0 with θ = 0 for appropriately chosen parameters k jwith N > 2, the model (9) can exhibit stable limit cycles that describe nonharmonic oscillatory behavior.
Importantly, in the fully dissipative case γ > 0 with θ > 0, the CD oscillator model (1) can be used to de-
termine the non-equilibrium thermodynamic variables FNE andUNL and the statistical entropy S of the
self-oscillator.
The variables FNE ,UNL , and S were estimated from experimental data in a three-step approach. First,
the model parameters of the Hamiltonian k1, . . . ,kn were determined using linear regression analysis [13,14]. Second, the canonical-dissipative parameters θ andB were estimated. To this end, we followed [8–10]
and calculated the Hamiltonian energy valuesH from equations (11–13) and the experimentally observed
time series x(t ) and v(t ) = dx/dt using the parameters estimates k1, . . . ,kn . Third, following again theearlier work [9, 10], we calculated FNE ,UNL , and S from equation (6) and the estimated values of θ and B .
A total of ten participants (undergraduate students from the University of Connecticut who received
partial course credit for their participation) were tested. Experimental procedures (as described above)
were approved by the University’s Institutional Review Board (IRB). Two participants performed oscil-
latory movements that were subjected to a great amount of jerk and could not be classified with the
aforementioned data analysis procedure as limit cycle oscillations. The data of those two participants
were discarded. Only the data from the remaining eight participants were evaluated. Two models were
tested: the baseline model involving only V0 and the parameters k1, k2 (for the harmonic oscillator case)and the N = 5model (for the nonharmonic oscillator case) involving parameters k1, . . . ,k6 and terms upto the order x6.
Table 1 shows descriptive measures (oscillation frequency and amplitude) of the observed oscillatory
movements for the three frequency conditions. As expected, oscillation frequency ω increased signif-
icantly across the frequency conditions [p <0.01, F(2.14)=43.683]. The amplitude A showed a U-shaped
pattern with theminimum at the comfortable condition. The effect of the frequency on A was statistically
significant [p <0.05, F(2.14)=5.045]. Table 2 shows the improvement ∆R2 of the R2 goodness of fit scores
when comparing the nonharmonic case with the harmonic case. By definition of the fitting procedure,
Table 1. Angular frequencies ω and oscillation amplitudes A as functions of the three frequency (speed)
conditions (slow, comfortable, fast). Angular frequencies were determined as peak frequencies ob-
served in the Fourier spectrum of the movement trajectories x(t ). Amplitudes were calculated from
the difference between averaged maximal and averaged minimal values of x(t ): A = [Peaks(MAX) −
Peaks(MIN)]/2. Standard errors in parenthesis.
Slow Comfortable Fast
ω [rad/s] 5.35 (0.36) 7.01 (0.47) 9.10 (0.71)
A [a.u.] 0.27 (0.03) 0.25 (0.03) 0.28 (0.03)
Table 2. Improvements of model fits quantified by R2-improvements scores ∆R2 observed for the three
experimental frequency (speed) conditions.
Slow Comfortable Fast
∆R2 h 1.6 2.6 1.4
34001-4
Thermodynamics of voluntary behavior
the R2 scores of the more comprehensive model (nonharmonic case) should exceed the R2 scores of the
less comprehensive model (harmonic case) such that ∆R2 > 0. The ∆R2 scores reported in table 2 reflect
this fundamental feature of regression models. For our purposes, table 2 demonstrates that we were able
to improve the fits of the self-oscillator models by taking nonharmonic contributions into account.
Table 3 presents the bifurcation parameter B as a function of frequency and model type. Consistent
with the previous studies [8–10], there was a statistically significant increase of B from slow to fast os-
cillations speeds [harmonic case: p <0.01, F(2.14)=11.722; nonharmonic case: p <0.01, F(2.14)=10.694].
That is, B increased with increasing oscillation frequency. Importantly, the same pattern was qualita-
tively observed for both models. Table 4 displays the statistical entropy S and the non-equilibrium ther-
modynamical variables UNL and FNL for the models and frequency conditions. Entropy S and internal
energyUNL scores increased significantly as functions of oscillation frequency [harmonic case: p <0.01,
F(2.14)=35.804 for S and p <0.05, F(2.14)=3.991 forUNL ; nonharmonic case: p <0.01, F(2.14)=36.582 for
S and p =0.06, F(2.14)=3.503 forUNL]. Note that the increase ofUNL in the nonharmonic case was onlymarginally statistically significant. Consistent with the previous studies [9, 10, 15], the non-equilibrium
free energy FNL decreased when oscillation frequency increased. The effect was marginally statisticallysignificant [harmonic case: p =0.07, F(2.14)=3.188; nonharmonic case: p =0.09, F(2.14)=2.810]. We will
return to this issue below.
Table 3. Bifurcation parameter B obtained for different frequency (speed) conditions as obtained from
the harmonic and nonharmonic model-based data analysis. Standard errors in parenthesis.
Case B (Slow) B (Comfortable) B (Fast)
[a.u.] [a.u.] [a.u.]
Harmonic 1.3 (0.3) 2.1 (0.6) 4.1 (1.1)
Nonharmonic 1.3 (0.3) 2.2 (0.7) 4.5 (1.3)
Table 4. Variables S, UNL , and FNL as functions of the frequency (speed) conditions (S=slow, C=comfor-table, F=fast) and model type. Standard errors in parenthesis.
Model- S(S) S(C) S(F) UNL(S) UNL(C) UNL(F) FNL(S) FNL(C) FNL(F)type [a.u.] [a.u.] [a.u.] [a.u.] [a.u.] [a.u.] [a.u.] [a.u.] [a.u.]
Harmonic 1.1 1.3 1.9 0.5 0.9 2.9 -1.1 -3.2 -14.6
(0.2) (0.2) (0.3) (0.1) (0.5) (1.4) (0.6) (2.1) (8.2)
Non- 1.1 1.3 1.8 0.4 0.8 2.8 -1.0 -2.8 -13.9
harmonic (0.2) (0.2) (0.3) (0.1) (0.4) (1.4) (0.5) (1.9) (8.1)
Our experimental design allowed us to examine and manipulate voluntary human performance in
an oscillatory motor control task without the use of an external pacing signal. Moreover, we were able to
account for nonharmonic oscillator contributions in the conservative part of the CDmodelling approach.
As expected, the nonharmonic models matched the data better than the harmonic models. Importantly,
the thermodynamic state variables showed for the harmonic and nonharmonic analyses the same kind
of patterns across the three frequency conditions. Therefore, it seems that (at least for the experimental
paradigm of the present study) the thermodynamic variables are not affected by the model type.
The pumping parameter B increased with oscillation frequency. It has been suggested that B mea-
sures the distance to the Hopf bifurcation point of the canonical-dissipative self-oscillator [9, 10]. Accord-
ingly, the non-equilibrium free energy decayed when the tested “human” self-oscillators operated further
away from their bifurcation points. It has been argued that this feature is consistent with the decay of the
Landau free energy of magnetic phase transitions [16] when the temperature is decreased further below
the critical transition temperature [9, 10].
For the sample of eight participants, the decay of the non-equilibrium free energywas onlymarginally
statistically significant. However, a detailed inspection of the individual participant data revealed that
two of the eight participants showed free energy scores FNL that were by a factor of 10 larger than thescores obtained for the remaining six participants. Analyzing the non-equilibrium free energy only for
34001-5
J.M. Gordon, S. Kim, T.D. Frank
the remaining “homogeneous” group of six participants, we found for FNL the same qualitative pattern asfor the whole group: the free energy decayed with an increasing oscillation frequency. However, for this
homogeneous subset of participants, the effect of speed was also statistically significant (p <0.05 for both
models). Future experimental work may be conducted to explore this participant effect in more detail.
Acknowledgements
Preparation of this manuscript was supported in part by National Science Foundation under the IN-
SPIRE track, grant BCS-SBE-1344275.
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Лiнiйна нерiвноважна термодинамiка людської поведiнки:
метод канонiчно-дисипативного рiвняння Фоккера-Планка з
потенцiалами,що включають ангармонiзми
Дж.М. Гордон1, С.Кiм1, Т.Д. Франк1,2
1 CESPA, факультет психологiї, Унiверситет Коннектикуту, Сторрс, CT 06269, США
2 Факультет фiзики, Унiверситет Коннектикуту, Сторрс, CT 06269, США
Представлено нову експериментальну парадигму та новий пiдхiд до моделювання роботи осциляторного
людського двигуна з допомогою ключового концепту фiзики конденсованої речовини, а саме— змiнних
термодинамiчного стану. Для цього в новiй експериментальнiй парадигмi учасники здiйснювали маятни-
ковi коливнi рухи з особисто вибраними частотами, що вiдмiнне вiд попереднiх дослiджень, де викори-
стовувалися синхронiзуючi сигнали. У новому пiдходiмоделювання використано канонiчно-дисипативну
границю циклiчно осциляцiйної моделi з консервативною частиною, що враховує негармонiчнi коливнi
компоненти— у цьому iнша вiдмiннiсть вiд попереднiх дослiджень, у яких розглядалися лише гармонiчнi
компоненти. В узгодженнi з теорiєю Ландау магнiтних фазових переходiв ми знайшли,що вiльна енергiя
осциляцiйної моделi зменшується, коли учасники пiдтримують коливання з вiддаленням вiд точки бiфур-
кацiї Хопфа для канонiчно-дисипативної границi циклiчного осцилятора.
Ключовi слова: фiзика життя, людська поведiнка, канонiчно-дисипативнi системи, змiннi
термодинамiчного стану
34001-6
http://dx.doi.org/10.1007/BF01391586
http://dx.doi.org/10.5488/CMP.13.13001
http://dx.doi.org/10.1016/j.biosystems.2015.01.002
http://dx.doi.org/10.1142/S1230161215500079
http://dx.doi.org/10.1140/epjst/e2012-01529-y
http://dx.doi.org/10.1142/S0217979211057712
http://dx.doi.org/10.1016/j.neulet.2007.09.066
http://dx.doi.org/10.1103/PhysRevE.74.051905
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