We study the influence of thermal fluctuations on the buckling behavior of thin elastic capsules with spherical rest shape. Above a critical uniform pressure, an elastic capsule becomes mechanically unstable and spontaneously buckles into a shape with an axisymmetric dimple. Thermal fluctuations affect the buckling instability by two mechanisms. On the one hand, thermal fluctuations can renormalize the capsule's elastic properties and its pressure because of anharmonic couplings between normal displacement modes of different wavelengths. This effectively lowers its critical buckling pressure [Košmrlj and Nelson, Phys. Rev. X 7, 011002 (2017)2160-330810.1103/PhysRevX.7.011002]. On the other hand, buckled shapes are energetically favorable already at pressures below the classical buckling pressure. At these pressures, however, buckling requires to overcome an energy barrier, which only vanishes at the critical buckling pressure. In the presence of thermal fluctuations, the capsule can spontaneously overcome an energy barrier of the order of the thermal energy by thermal activation already at pressures below the critical buckling pressure. We revisit parameter renormalization by thermal fluctuations and formulate a buckling criterion based on scale-dependent renormalized parameters to obtain a temperature-dependent critical buckling pressure. Then we quantify the pressure-dependent energy barrier for buckling below the critical buckling pressure using numerical energy minimization and analytical arguments. This allows us to obtain the temperature-dependent critical pressure for buckling by thermal activation over this energy barrier. Remarkably, both parameter renormalization and thermal activation lead to the same parameter dependence of the critical buckling pressure on temperature, capsule radius and thickness, and Young's modulus. Finally, we study the combined effect of parameter renormalization and thermal activation by using renormalized parameters for the energy barrier in thermal activation to obtain our final result for the temperature-dependent critical pressure, which is significantly below the results if only parameter renormalization or only thermal activation is considered.
We study the axisymmetric response of a complete spherical shell under homogeneous compressive pressure p to an additional point force. For a pressure p below the classical critical buckling pressure pc, indentation by a point force does not lead to spontaneous buckling but an energy barrier has to be overcome. The states at the maximum of the energy barrier represent a subcritical branch of unstable stationary points, which are the transition states to a snap-through buckled state. Starting from nonlinear shallow shell theory we obtain a closed analytical expression for the energy barrier height, which facilitates its effective numerical evaluation as a function of pressure by continuation techniques. We find a clear crossover between two regimes: For p/pc 1 the post-buckling barrier state is a mirror-inverted Pogorelov dimple, and for (1 − p/pc) 1 the barrier state is a shallow dimple with indentations smaller than shell thickness and exhibits extended oscillations, which are well described by linear response. We find systematic expansions of the nonlinear shallow shell equations about the Pogorelov mirror-inverted dimple for p/pc 1 and the linear response state for (1 − p/pc) 1, which enable us to derive asymptotic analytical results for the energy barrier landscape in both regimes. Upon approaching the buckling bifurcation at pc from below, we find a softening of an ideal spherical shell. The stiffness for the linear response to point forces vanishes ∝ (1 − p/pc) 1/2 ; the buckling energy barrier vanishes ∝ (1 − p/pc) 3/2 ; and the shell indentation in the barrier state vanishes ∝ (1 − p/pc) 1/2 . This makes shells sensitive to imperfections which can strongly reduce pc in an avoided buckling bifurcation. We find the same softening scaling in the vicinity of the reduced critical buckling pressure also in the presence of imperfections. We can also show that the effect of axisymmetric imperfections on the buckling instability is identical to the effect of a point force that is preindenting the shell. In the Pogorelov limit, the energy barrier maximum diverges ∝ (p/pc) −3 and the corresponding indentation diverges ∝ (p/pc) −2 . Numerical prefactors for proportionalities both in the softening and the Pogorelov regime are calculated analytically. This also enables us to obtain results for the critical unbuckling pressure and the Maxwell pressure.
The "edge of chaos" phase transition in artificial neural networks is of renewed interest in light of recent evidence for criticality in brain dynamics. Statistical mechanics traditionally studied this transition with connectivity k as the control parameter and an exactly balanced excitation-inhibition ratio. While critical connectivity has been found to be low in these model systems, typically around k = 2, which is unrealistic for natural neural systems, a recent study utilizing the excitationinhibition ratio as the control parameter found a new, nearly degree independent, critical point when connectivity is large. However, the new phase transition is accompanied by an unnaturally high level of activity in the network.Here we study random neural networks with the additional properties of (i) a high clustering coefficient and (ii) neurons that are solely either excitatory or inhibitory, a prominent property of natural neurons. As a result we observe an additional critical point for networks with large connectivity, regardless of degree distribution, which exhibits low activity levels that compare well with neuronal brain networks. arXiv:1903.12632v2 [cond-mat.dis-nn]
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