critical computing critical cracking con critical energy react critical escape altit critical field critical friend critical linguistics critical nucleus critical properties critical rate of run- critical reviewstm in critical sampling rat critical social work critical sociology critical witness critical-- crit...
with explicit constantsC_Ndepending only onN. Note that, as a reflection of the Brézis–Nirenberg phenomenon,Venters pointwisely into the asymptotic coefficient in (1.7) and (1.8), while it enters non-locally throughQ_Vinto the asymptotic coefficient in Theorem1.3. Asymptotics (1.7) and (1.8)...
Both the Fokker–Planck and path-integral formulations inherit their functional form from assumptions about the statistics of random fluctuations in (1). For example, the most likely path—or path of least action—is the path taken when the fluctuations take their most likely value of zero. This...
Therefore what we noticed in a special case turns out to be true in general. (What if F does not come from a potential? Then (40.2) has no solution at all. Energy can be generated, or lost by the atoms running around in cyclic paths for which the work done is not zero, and no ...
[3, p. xii]. This work remarkably expanded our understanding of the pivotal role Feynman graphs currently hold in high-energy physics. However, despite this existing literature, there are still notable gaps in historical investigations regarding an essential technical aspect related to the application...
As the model does not have a free energy defined in classical terms, we resort to a dynamical equivalent in the form of a generating functional. We apply a path integral approach to obtain exact solutions on its statistical moments and nonequilibrium thermodynamic properties. Unlike the replica ...
where the integral is over the primitive cell of the lattice. A Kerr nonlinearity would give Δε∼|E→|2 material absorption would produce an imaginary frequency correction (decay coefficient) from a small imaginary Δε and so on. Similarly, one can compute the shift in frequency from a ...
We consider two interacting systems when one is treated classically while the other system remains quantum. Consistent dynamics of this coupling has been shown to exist, and explored in the context of treating space-time classically. Here, we prove that
In short, the brain seeks to minimise free energy, which is mathematically equivalent to maximising model evidence. This view of neuronal responses can be conceived with respect to Hamilton’s principle of least action, whereby action is the path integral of free energy. ...
where μ is the reduced mass of the two particles, and E is the kinematic energy of the two-particle system. Obviously, the above integral does not diverge in D=4. Using the minimal subtraction (MS) scheme which subtracts the 1/(D−4) pole before taking the D→4 limit, one obtains...