Equipartition Theorem
Violation
“Flying Ice Cube”
Ladislav Bartoš
March 8, 2021
FB810/C9926 Course
Mar 8, 2021 Ladislav Bartoš
Numerical Integrators
2
●
solved by numerical integration => provides trajectories
●
e.g. Verlet algorithm, leap-frog algorithm
●
integrators keep the total energy constant (NVE)
microcanonical
ensemble
Newton equations of motion
Mar 8, 2021 Ladislav Bartoš
Temperature Coupling
3
●
to achieve constant temperature (NVT), we must
employ a thermostat
●
temperature <= kinetic energy <= particles velocity
●
constant temperature <= constant average velocity
●
periodically, we must:
a) generate brand new velocities, or
b) rescale the current velocities
canonical ensemble
& =>
Mar 8, 2021 Ladislav Bartoš
Berendsen Thermostat
4
●
every time-step, velocity of each particle is multiplied
by a rescaling factor λ
●
current temperature gradually changes and converges
to the target temperature
time-step
coupling
constant
target temperature
current temperature
rescaling
factor target temperature will
be reached in 1 step
target temperature will
never be reached
Mar 8, 2021 Ladislav Bartoš
Flying Ice Cube
5
vib vib+trans TOTAL
vib vib+trans TOTAL
●
kinetic energy is the same in both cases
●
the distribution of kinetic energy differs
“flying ice cube”
the system effectively freezes
Mar 8, 2021 Ladislav Bartoš
Equipartition Theorem
6
●
“real systems usually do not behave as flying ice cubes”
●
equipartition theorem is violated
Equipartition theorem
Each degree of freedom
has the same average
kinetic energy.
each direction of translation
each axis of rotation
each mode of vibration
Berendsen
thermostat
translation of the center of mass
rotation about the center of mass
kinetic energy
kinetic
energy
Mar 8, 2021 Ladislav Bartoš
Cause of the Problem
7
●
why does Berendsen thermostat remove some motions?
●
it tries to achieve constant instantaneous kinetic energy
=> fluctuations of Ekin are suppressed
●
NVT ensemble does allow the kinetic energy to fluctuate!
●
BT samples some vaguely defined ensemble, not NVT
“Suppressed Ekin fluctuations force the thermostat to
convert Ekin associated with high frequency motions to
Ekin associated with low frequency motions.”
– Math, unknown date
Mar 8, 2021 Ladislav Bartoš
Solutions
8
●
use Monte Carlo instead of Molecular Dynamics
●
remove the center of mass motions
comm-mode = Linear / Angular
●
use very long coupling time
●
periodically reassign velocities
●
use Andersen thermostat
●
use a proper NVT velocity-rescaling thermostat
a combination
of these three
historical
solutions
Mar 8, 2021 Ladislav Bartoš
Velocity-Rescaling Thermostat
●
works similarly to the Berendsen thermostat
●
but adds a random (stochastic) factor to the
rescaling factor λ
●
this allows for the fluctuations of kinetic energy
=> canonical ensemble is generated (sampled)
=> equipartition theorem is not violated
=> no flying ice cube effect
9
Mar 8, 2021 Ladislav Bartoš
Take-Home Message
10
Do not use Berendsen thermostat!
tcoupl = berendsen
tcoupl = v-rescale
tcoupl = nose-hoover
tcoupl = andersen
Gromacs
Mar 8, 2021 Ladislav Bartoš
Disclaimer
11
●
system may freeze also for other reasons
●
e.g. Martini CG water tends to freeze when generated
on a lattice (grid)
●
using the right thermostat does not solve everything!