Vibrational thermodynamic instability of recursive networks
Dipartimento di Fisica and INFM,
Università di Parma Parco Area delle Scienze 7/A, 43100 Parma,
2 Dipartimento di Fisica and INFM, UdR and SMC Università di Roma “La Sapienza” - Piazzale A. Moro 2, 00185 Roma, Italy
Accepted: 25 March 2002
In this letter we study the thermodynamic stability problem for a generic geometrical structure by considering the harmonic vibrational dynamics of a network of masses and springs. We relate the stability properties of the network to the recurrence properties of random walks or, equivalently, to the vibrational spectral dimension . This is an extension of the Peierls theorem for the thermodynamic instability of low-dimensional crystalline structures, proving that stability is possible if and only if . We predict the existence of an instability critical length on structurally disordered materials. Our results are discussed on the specific case of a Sierpinki-gasket fractal, which is exactly solvable.
PACS: 05.50.+q – Lattice theory and statistics (Ising, Potts, etc.) / 61.43.-j – Disordered solids / 65.60.+a – Thermal properties of amorphous solids and glasses: heat capacity, thermal expansion, etc
© EDP Sciences, 2002