Biomedical Engineering Reference
In-Depth Information
individual or a subcompartment), the local velocity, and not the acceleration,
is proportional to the local force. This relation is called overdamped force-
velocity response, and it is characteristic of other IBMs [111, 113]:
1
k
x
2
F
x
2
=
x
2
v
x
2
=
v
x
2
:
(4.13)
The coecient k
x
2
(t) is the net rate of transition of site x, the difference be-
tween its probability of moving and staying still, P((x) ! (x
0
))P((x)9
(x
0
)), as in [165]. As analytically demonstrated again in [165] and in [254]
for a particular case, k
x
2
(t) is related to the Boltzmann temperature. We
can therefore write
F
x
2
/
1
v
x
2
:
(4.14)
T
As far as the above-mentioned published results are concerned, we here pre-
fer to use a proportional dependence between k
x
2
(t) and T
(
x
)
(t) and not
an equation, since the exact relation between the Monte Carlo spin copy at-
tempts and the continuous time, as well as the kinematics application of the
Metropolis-like algorithm, are still debated and are a persistent source of crit-
icism.
Using (4.14) and assuming, for the sake of simplicity, the proportionality
coecient constant throughout the object, (4.12) can be rewritten as
t
/
X
x
2
X
H
T
v
2
x
2
=
1
1
v
2
x
2
;
(4.15)
T
x
2
given that T
is a global property of the entire object. Let us now decompose
the velocity v
x
2
as
v
x
2
= v
CM
+ w
x
2
;
(4.16)
where v
CM
is the velocity of the object center of mass (defined in Equation
(1.11)) and w
x
2
a local fluctuation. Simple calculations lead to
X
v
x
2
=
X
x
2
+
X
x
2
+
X
x
2
v
CM
w
x
2
= v
CM
a
volume
w
x
2
; (4.17)
x
2
where the second term of the sum vanishes. Therefore, we obtain
X
+
X
x
2
v
2
x
2
= (v
CM
)
2
a
volume
w
2
x
2
;
(4.18)
x
2
that, substituted in (4.12), leads to
"
#
+
X
x
2
H
t
/
1
(v
CM
)
2
a
volume
w
2
x
2
:
(4.19)
T
Given that
H
t
=
H
x
CM
v
CM
;
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