As a first step, we analyze discontinuous changes by considering
the bullet-wood non-elastic collision illustrated
in Fig. 1. Conservation of momentum
determines that after collision both masses have a common velocity,
. The
amount of energy contained in the system before the collision is
, and, the amount of energy in the
system after the collision
is 
This implies that the collision causes a loss of energy equal to
.
Imposition of conservation of momentum results in a
discontinuous loss of energy to the environment, and is represented
by a Dirac pulse
whose area is determined by the model.
The explanation for this energy loss is that an instantaneous change in
system configuration results in two independent buffers (the masses)
becoming dependent, which then causes dissipation of energy to the
environment as heat.
In case the environment is considered isothermal, this thermal
energy flow need not be explicitly modeled
by a heat sink between the system and the environment
(like dissipation of resistive elements).
Note that the instantaneous loss of energy would not occur if a resistive
element modeled material deformation between the buffers
(i.e., their connection was non-ideal).
From a physical perspective, buffers that become dependent have to be analyzed carefully. Bond graph modeling theory assumes that physical system behaviors change slowly enough to satisfy the lumped parameter assumption where distribution of energy within energy buffers is considered homogeneous. Therefore, phenomena like dynamic turbulence effects caused by sudden large pressure differences in a tank cannot be easily taken into account in system models. The observation of an instantaneous energy redistribution when buffers become dependent is basically an artifact of an abstract model operating on a coarse time scale.
To study the lumped parameter assumption in detail, consider the free expansion experiment conducted by Gay-Lussac and Joule shown in Fig. 4 [3]. A chamber is made up of two connected bulbs with an on-off valve that switches the connection on or off. Initially, only the left bulb contains a gas and the valve is closed. When the valve connecting the two volumes is opened, the gas in the left bulb expands freely and starts diffusing into the right bulb. Even if the connecting orifice is non-resistive this diffusion introduces turbulence effects that result in non-homogeneities, which violate the lumped parameter assumption. From a thermodynamics perspective, when the goal is to compute equilibrium temperatures and pressures, using energy balance on a homogeneous model is adequate.
For the dynamic effects to be negligible in the time scale of interest, they have to occur in a mode of continuous operation. In case of the above experiment, if the valve were closed quickly enough after opening, i.e., we have two instantaneous changes, the gas cannot diffuse and the homogeneous distribution over both of the volumes is never actually established. For CSPEC conditions based on the energy variables involved, the time scale is too small for the lumped parameter assumption to hold, and bond graphs cannot be used to model such processes. In effect, an immediate closing of the opening leads to no redistribution of energy which substantiates the observation that there never was a connection in real time.
Figure 4: Instantaneous free expansion of a gas by diffusion.
In conclusion, switching conditions based on energy stored in buffers that are between dependent and independent without an intervening mode of continuous operation are inconsistent with bond graph assumptions and, therefore, prohibited. In this case, either model refinement or another modeling approach has to be chosen.