TY - JOUR

T1 - Glassy dynamics from generalized mode-coupling theory

T2 - Existence and uniqueness of solutions for hierarchically coupled integro-differential equations

AU - Biezemans, Rutger A.

AU - Ciarella, Simone

AU - Çaylak, Onur

AU - Baumeier, Björn

AU - Janssen, Liesbeth M.C.

PY - 2020/10

Y1 - 2020/10

N2 - Generalized mode-coupling theory (GMCT) is a first-principlesbased and systematically correctable framework to predict the complex relaxation dynamics of glass-forming materials. The formal theory amounts to a hierarchy of infinitely many coupled integro-differential equations, which may be approximated using a suitable finite-order closure relation. Although previous studies have suggested that finite-order GMCT leads to well-defined solutions, and that the hierarchy converges as the closure level increases, no rigorous and general result in this direction is known. Here we unambiguously establish the existence and uniqueness of solutions to generic, schematic GMCT hierarchies that are closed at arbitrary finite order. We consider two types of commonly invoked closure approximations, namely mean-field and exponential closures. We also distinguish explicitly between overdamped and underdamped glassy dynamics, corresponding to hierarchies of first-order and second-order integro-differential equations, respectively. We find that truncated GMCT hierarchies closed under an exponential closure conform to previously developed mathematical theories, both in the overdamped and underdamped case, such that the existence of a unique solution can be readily inferred. Self-consistent mean-field closures, however, of which the well-known standard-MCT closure approximation is a special case, warrant additional arguments for mathematical rigor. We demonstrate that the existence of a priori bounds on the solution is sufficient to also prove that unique solutions exist for such self-consistent hierarchies. To complete our analysis, we present simple arguments to show that these a priori bounds must exist, motivated by the physical interpretation of the GMCT solutions as density correlation functions. Overall, our work contributes to the theoretical justification of GMCT for studies of the glass transition, placing this hierarchical framework on a firmer mathematical footing.

AB - Generalized mode-coupling theory (GMCT) is a first-principlesbased and systematically correctable framework to predict the complex relaxation dynamics of glass-forming materials. The formal theory amounts to a hierarchy of infinitely many coupled integro-differential equations, which may be approximated using a suitable finite-order closure relation. Although previous studies have suggested that finite-order GMCT leads to well-defined solutions, and that the hierarchy converges as the closure level increases, no rigorous and general result in this direction is known. Here we unambiguously establish the existence and uniqueness of solutions to generic, schematic GMCT hierarchies that are closed at arbitrary finite order. We consider two types of commonly invoked closure approximations, namely mean-field and exponential closures. We also distinguish explicitly between overdamped and underdamped glassy dynamics, corresponding to hierarchies of first-order and second-order integro-differential equations, respectively. We find that truncated GMCT hierarchies closed under an exponential closure conform to previously developed mathematical theories, both in the overdamped and underdamped case, such that the existence of a unique solution can be readily inferred. Self-consistent mean-field closures, however, of which the well-known standard-MCT closure approximation is a special case, warrant additional arguments for mathematical rigor. We demonstrate that the existence of a priori bounds on the solution is sufficient to also prove that unique solutions exist for such self-consistent hierarchies. To complete our analysis, we present simple arguments to show that these a priori bounds must exist, motivated by the physical interpretation of the GMCT solutions as density correlation functions. Overall, our work contributes to the theoretical justification of GMCT for studies of the glass transition, placing this hierarchical framework on a firmer mathematical footing.

KW - Aging

KW - Exact results

KW - Glassy dynamics

KW - Mode coupling theory

KW - Slow relaxation

UR - http://www.scopus.com/inward/record.url?scp=85096073278&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/abb6e1

DO - 10.1088/1742-5468/abb6e1

M3 - Article

AN - SCOPUS:85096073278

VL - 2020

JO - Journal of Statistical Mechanics : Theory and Experiment

JF - Journal of Statistical Mechanics : Theory and Experiment

SN - 1742-5468

IS - 10

M1 - 103301

ER -