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[EQUATION] The amplitude is set by the microscopically-determined mean cage expansion length, [MATH] [EQUATION] where [MATH] and grows with density or cooling. The prefactor of 3/32 in Eq.( ) follows from assuming each spherical particle in the cage independently hops in a random direction by [MATH] |
There are two ways to then compute the elastic barrier. One could invoke literal continuum mechanics, as done by Dyre in his seminal phenomenological approach . However, in ECNLE theory the local cage and long range collective elastic aspects are intimately related. Given the former is described microscopically, for co... |
[EQUATION] where [MATH] is relative to the cage center and [MATH] . Note the long range nature of the integrand in eq 3 which decays as [MATH] , and hence the total elastic barrier converges slowly to its full value with the leading correction scaling as [MATH] |
The sum of the coupled (and in general temperature and density dependent) local and elastic collective barriers determine the mean total barrier for the alpha relaxation process: |
[EQUATION] The elastic barrier increases much more strongly with increasing density or cooling than its cage analog, and dominates the growth of the alpha time as the laboratory glass transition is approached . A generic measure of the average structural relaxation time follows from a Kramers calculation of the mean fi... |
[EQUATION] where [MATH] is the absolute magnitude of the barrier curvature in units of [MATH] . The alpha time is expressed in units of a ”short time/length scale” relaxation process (cage-renormalized Enskog theory) the explicit formula for which is given elsewhere . Physically, it is meant to capture the alpha proces... |
II.2 Mappings for Molecular and Polymeric Liquids The theory is rendered quantitatively predictive for rigid molecular liquids via a mapping to an effective hard sphere fluid guided by the requirement that it exactly reproduces the equilibrium dimensionless density fluctuation amplitude (compressibility) of the liquid ... |
[EQUATION] The first equality employs Percus-Yevick (PY) integral equation theory for hard sphere fluids. The final equality is an accurate analytic description of experimental data derived previously . Temperature enters all 3 factors in [MATH] . This mapping determines a material-specific, temperature-dependent effec... |
[EQUATION] Thus, in practice, 4 known chemically-specific parameters enter in the minimalist mapping : [MATH] and [MATH] (interaction site level entropic and cohesive energy EOS parameters, respectively), the number of elementary sites that define a rigid molecule, [MATH] (e.g., [MATH] for benzene), and hard sphere dia... |
Figure 1 shows mean alpha relaxation time calculations for orthoterphenyl (OTP) in two temperature representations; for this system no adjustable parameter agreement with experiment has been documented . Detailed analytic and numerical analyzes of the theoretical form of the temperature dependence of the alpha time hav... |
Polymers have additional complexities associated with conformational isomerism and chain connectivity. As a minimalist model the polymer liquid is replaced by a fluid of disconnected Kuhn-sized segments modeled as non-interpenetrating hard spheres composed of a known number of interaction sites, [MATH] , and effective ... |
In this article, we present representative calculations for a subset of organic molecules and polymer melts previously studied . Specifically , polystyrene (PS; fragility = [MATH] ) and orthoterphenyl (fragility [MATH] ) where [MATH] , very high fragility ( [MATH] ) polycarbonate (PC) where [MATH] , and low fragility (... |
III Temperature Dependence of Short and Long Time Dynamics and Effective Volume Fraction III.1 Apparent Plateau Mean Square Displacement |
The single particle mean square displacement (MSD) at intermediate time scales where particles are approximately ”transiently localized” is a quantity of interest in simulation and experiment (e.g., quasi-elastic neutron scattering ). In a log-log plot, the displacement corresponding to the minimum non-Fickian slope of... |
[EQUATION] This practical measure of a dynamic localization length is not the same as the literal minimum of the dynamic free energy at [MATH] |
Calculations of [MATH] as a function of temperature for several systems are shown in Fig.2. Possible universality based on a high temperature crossover temperature, [MATH] , is explored where the latter is defined via when the total barrier is either 1 or 3 [MATH] . The doubly normalized plot in Fig.2 reveals that over... |
[EQUATION] These results can be potentially tested against simulation and experiment. Note that while the numerical data in Figure 2 can be reasonably described as linear in temperature over the narrow range probed in simulation, the functional form is nonlinear at the lower temperatures of primary experimental interes... |
III.2 Dynamic Barriers Fundamental connections of the alpha relaxation time and measures of short time dynamics have been predicted by ECNLE theory in prior studies . Recently, Simmons suggested based on simulations performed at relatively high temperatures that a roughly exponential, but non-universal, connection exis... |
Motivated by the above, Figure 3 plots our calculations for the temperature-dependent total barrier divided by its value at [MATH] against the normalized square of [MATH] . For PS, PC, and OTP (not plotted, identical to PS) they are well fit (including all of the deeply supercooled regime) by: |
[EQUATION] where [MATH] [MATH] [MATH] are positive system-specific constants, and c increases monotonically with fragility. Thus ECNLE theory does predict a specific exponential connection between the barrier and [MATH] if expressed in a dimensionless form. Note that for the very low fragility PIB ( [MATH] ), the plot ... |
Various theoretical models based on different physics generally correspond to different forms of the temperature dependence of the effective barrier . Mauro et al have proposed a phenomenological model they claim can fit experimental data over many decades based on a configurational entropy perspective significantly mo... |
With the above motivation, the main frame of Figure 4a plots ECNLE theory barrier calculations for OTP in a log-linear inverse temperature Angell representation. The individual contributions to the total barrier are not very exponential in inverse temperature. However, surprisingly, the total barrier over a wide range ... |
Figure 4b plots the same OTP results in the less common linear in temperature format. Curiously, the elastic and total barriers are reasonably exponential in this representation. The inset of Fig.4b shows the corresponding value of [MATH] for OTP, which is also roughly exponential. This behavior has physical meaning gi... |
III.3 Mapped Volume Fraction and Dynamic Crossovers The key quantity to treat thermal liquids in ECNLE theory is the effective hard sphere temperature-dependent volume fraction of Eq.(7). Figure 5a shows calculations of this quantity for the four systems of present interest in the standard inverse temperature represent... |
[EQUATION] where [MATH] and [MATH] depend on material. This implies that if the quantities that enter Eq. ( ) are expanded through linear order in [MATH] , then the content of the mapping is almost fully captured. We note the slope for OTP in Fig.5b is [MATH] |
[MATH] , nearly identical to its linear expansion coefficient of [MATH] [MATH] . Precise agreement should not be expected since the mapping is based on the dimensionless compressibility which has 3 temperature dependent quantities. On the other hand, the naive idea that under the isobaric (1 atm) conditions of interest... |
The implications of Eq.( 11 ) for dynamics are interesting. First note that in the standard representation of Fig.5a the smooth curve could be crudely viewed as consisting of high and low temperature linear branches. With such a construction (not shown), for PS we find the lines intersect at [MATH] |
[MATH] , corresponding to [MATH] . A similar exercise for OTP and PC yields [MATH] and 1.3, respectively. The absolute value of [MATH] , and its reduction with increasing fragility, agrees well with trends of experimentally-deduced dynamical crossover temperatures . The latter are based on empirically fitting the ideal... |
[MATH] , nearly identical to the [MATH] value found from Fig.5a. A caveat is that although dynamic properties plotted versus [MATH] show the crossover, the effective volume fraction of Fig.5b does not. |
IV Measures of Cooperativity At present, ECNLE theory focuses on average dynamic properties. Explicit space-time dynamic heterogeneity (DH) is not addressed. However, the concept of ”cooperativity” is not the same as DH. It can be analyzed in the ECNLE theory framework. |
IV.1 Cooperative Displacement Real space analyses of simulations have attempted to identify the number of particles involved in an relaxation event and, even more objectively, the total particle mean square displacement associated with a re-arrangement, defined here as [MATH] . Schall, Spaepen and Weitz experimentally ... |
Based on the coupled local-nonlocal physical picture of alpha relaxation in ECNLE theory, the ”number of re-arranging particles” is ill-defined (in contrast to other models such as Adams-Gibbs and RFOT which involve compact clusters). However, we can compute [MATH] . The cage consists of a central particle plus [MATH] ... |
[EQUATION] The collective elastic fluctuation contribution corresponds to a total displacement of: [EQUATION] This is far smaller than the local hopping contribution. Hence, the total linear displacement [MATH] |
[MATH] particle diameters, grows weakly with cooling, and is dominated by local physics even though the long range elastic effects make a large contribution to the activation barrier. The obtained modest value of [MATH] does seem reasonable compared to simulation studies . |
IV.2 Cooperativity Length Scale Since collective elastic effects involve a scale-free displacement field, there is no intrinsic length scale in the usual sense. However, a cooperativity length can be defined by asking a question recently explored in studies of thin film heterogeneous dynamics . There, one can define a ... |
[EQUATION] Note the slow inverse in distance decay to its asymptotic value. From this, a cooperativity length scale is defined as when a fixed percentage ( [MATH] ) of the bulk alpha time is recovered: |
[EQUATION] where [MATH] is proportional to the bulk elastic barrier. Prior work argued fragility is dominated by collective elasticity which is the origin of ”cooperativity” in ECNLE theory. From Eq.(15) one can write: |
[EQUATION] Figure 7 shows sample calculations of bulk for PS and OTP (they are almost identical) based on the criteria [MATH] and [MATH] . This cooperativity length grows strongly with cooling, and is well described by a cubic polynomial. For the 50 [MATH] criterion, bulk [MATH] at the laboratory [MATH] . The inset of ... |
[MATH] s) . Importantly, it is essentially the longest time scale that has been probed in molecular dynamics (MD) simulation. Hence, since existing MD simulations cannot access the deeply supercooled regime where the collective elastic effects become dominant, the molecular cooperativity lengths they can probe are mode... |
IV.3 Time-Length Scale Connection One can ask if a simple connection exists between the cooperativity length and alpha time, or its natural logarithm which defines an effective barrier. This question is of prime interest in diverse glass physics theories . Each theory typically has a (growing) length scale, with a dist... |
[EQUATION] Whether ECNLE theory obeys any of the above three relations is not a priori obvious given the many different microscopic quantities that enter the alpha time calculation and the presence of two barriers with distinct density and temperatures dependences. |
We have explored the above question for several thermal liquids and choices of the criterion parameter C. Remarkably, we generically find that all forms in Eq.(17) can represent extremely well our results for the alpha time, typically over 12-15 orders of magnitude in time. Figure 8 shows representative results for the... |
Alternative Continuum Mechanics Calculation of the Elastic Barrier V.1 Bulk Analysis and Comparison to Einstein Model Analog The original motivation for extending the local NLE theory of hopping to include collective elastic effects was the phenomenological ”shoving model” of Dyre . He derived the displacement field in... |
[EQUATION] where [MATH] is the high frequency dynamic shear modulus. The strain energy, identified as the elastic barrier, is then: |
[EQUATION] This basic form is similar to Eq. ( ) with three differences: (i) numerical prefactor, (ii) the macroscopic shear modulus replaces the single particle spring constant [MATH] , and (iii) the integrand decays not as [MATH] as does the molecular Einstein model, but much more quickly as [MATH] |
To establish the consequences of the above differences for bulk relaxation, we adopt an accurate analytic formula for G derived in prior NLE theory studies : |
[EQUATION] Substituting Eq.( 20 ) into Eq.( 19 ) gives [EQUATION] Hence, almost identical results per the molecular Einstein approach are obtained to within a nearly constant numerical prefactor. For the bulk relaxation time and [MATH] there are no conceptual differences between using continuum mechanics versus molecul... |
V.2 Cooperativity Length Scale Given the elastic energy decays much faster ( [MATH] ) in the continuum mechanics approach compared to the molecular Einstein analog ( [MATH] ), there must be significant differences for [MATH] . Figure 9 presents representative calculations analogous to those in Fig.7. One sees a massive... |
[EQUATION] Simple algebra yields the relation between the cooperativity lengths based on the two calculations labeled with subscripts [MATH] and [MATH] for continuum and Einstein, respectively: |
[EQUATION] The cube root relation explains the huge length scale reduction. Given the cubic polynomial fit in Fig. 7, it also explains to zeroth order the nearly inverse temperature dependence in Fig. 9. Note that based on the continuum mechanics calculation, the weakly varying with temperature local barrier now also a... |
V.3 Time-Length Scale Connection We have carried out the same numerical exercise as in section IVC to explore the validity of the three forms of the barrier-alpha time relationships of Eq.( 17 ). Results analogous to Fig.8 are shown in Fig.10. Remarkably good straight lines are again obtained, with a much larger appare... |
We conclude that the existence of a tight connection between the alpha time and a growing cooperativity length scale in ECNLE theory is present regardless of the approach used to compute the elastic barrier. However, the absolute magnitude and temperature dependence of the cooperativity length scale, and the apparent e... |
VI Discussion We have analyzed new aspects of ECNLE theory to provide deeper insight and address new questions. Calculations have been performed for the hard sphere fluid and thermal molecular and polymeric liquids of diverse fragilities. We find a near universality of the temperature-dependence of the apparent dynamic... |
The particle-level total displacement associated with the alpha event is found to be weakly temperature-dependent (grows with cooling) and only [MATH] particle diameters. An alternative amplitude-based criterion for determining a cooperativity length scale was also analyzed. It grows strongly with cooling, reaches very... |
The issue of the molecular Einstein versus literal continuum mechanics approach to computing the collective elastic barrier might be more incisively probed by performing new simulations and/or confocal imaging experiments in colloidal materials. This question is especially germane to how solid or vapor boundaries can ”... |
Acknowledgements. This work was performed at the University of Illinois and supported by DOE-BES under Grant No. DE-FG02-07ER46471 administered through the Frederick Seitz Materials Research Laboratory. We thank Professor David Simmons for many stimulating and informative discussions. |
# Source: arxiv 1806.05359 # Title: Convergence of Learning Dynamics in Information Retrieval Games # Sections: all # Downloaded: 2026-03-02T09:22:02.039582+00:00 |
Convergence of Learning Dynamics in Information Retrieval Games Abstract We consider a game-theoretic model of information retrieval with strategic authors. We examine two different utility schemes: authors who aim at maximizing exposure and authors who want to maximize active selection of their content (i.e., the numb... |
Introduction Information retrieval is probably the most central task carried out by consumers and users of on-line media. The basic information retrieval task involves ranking documents in a corpus by their relevance to the information needs expressed in a query. In adversarial retrieval settings such as the Web, infor... |
Despite the tremendous amount of work on information retrieval and SEO published during past decades, mathematical modeling of the aforementioned strategic behavior has only been formally suggested and studied recently ( ). One central question in this regard is whether learning dynamics, whereby at every step one auth... |
In this paper we introduce what is, to the best of our knowledge, the first attempt to explore the learning dynamics of strategic behavior in information retrieval systems such as the Web, through a formal theoretical model. Our main result proves that under the PRP, any better-response learning dynamics converges to a... |
1.1 Related Work The concept of mediators in strategic environments is widely known to the game-theory community ( ), and the design of a mediator (or in a different terminology, a mechanism) is often called mechanism design ( ). In the context of information retrieval, a search engine can be viewed as a mediator betwe... |
Considering strategic behavior in an information retrieval context is the aim of ). The work of presents a game-theoretic approach to information retrieval, and illustrates that the myopic static view falls short in dynamic and adversarial settings. explicitly assume that users will select the highest ranked result, a ... |
) consider mediator design in recommendation systems with strategic content providers. They highlight several fairness-related properties that a mediator should arguably satisfy, along with the requirement of pure Nash equilibrium existence. They claim against PRP, as they show that in their mathematical model the PRP ... |
Designing a mediator for improved social welfare was recently proposed by ), who also make the connection between recommendation systems and facility location games ( ). In their model as well, matching users with their nearest facility may yield a low social welfare in case the content providers are strategic. Their g... |
In this work, however, we do not study the social welfare, but rather focus on the learning dynamics . Learning dynamics is an important concept in machine learning and game theory ( ), and work on learning dynamics in games is considered instrumental, e.g., to understanding ad auctions ( ). Better-response learning dy... |
Another interesting class of games which are not potential games for which better-response dynamics always converge is ( ). However, that setting is quite remote from ours, as in ’s work the players share a common set of strategies. |
1.2 Our Contribution Our main conceptual contribution is the explicit analysis of learning dynamics in information retrieval systems that is motivated by strategic behavior. Our demonstration of convergence serves as an important justification for the use of the PRP, and should be taken into account when designing stab... |
The key technical contribution of this paper is the proof that under PRP any better-response dynamics converges to a pure Nash equilibrium. We prove this claim for both exposure-targeted and action-targeted utility schemes. As stated above, the convergence of better-response learning dynamics in our setting is obtained... |
1.3 Paper Organization The rest of the paper is organized as follows. Section formalizes the model we adopt, as well as an informal introduction to the relevant core game-theoretic concepts and an illustrative example. In Section we analyze better-response learning with the PRP mediator for both utility schemes. In Sec... |
Problem Statement An authors game is composed of a set of authors [MATH] , each owning one document/website/blog. [MATH] is the set of topics , and we assume both [MATH] and [MATH] are finite. An author’s pure strategy space is the set of all topics, i.e., she can choose to write her document on any topic. We further a... |
The matrix [MATH] is the quality matrix , where [MATH] represents the quality for author [MATH] ’s document if she decides to write on topic [MATH] . This modeling allows an author to have remarkable aptitude for one topic and poor aptitude for another. For example, an economic guru is able to write about sports, but h... |
The function [MATH] is the mediator , which plays the role of a ranking function or a search engine. The mediator ranks the documents selected by the authors w.r.t. a given query (or equivalently, a topic). We assume for simplicity that users always read the document ranked first. This assumption is consistent with man... |
The last component [MATH] is the utility function , which maps every strategy profile to a real-valued vector of length [MATH] . In this paper, we consider two different utility functions which are motivated by current applications. |
Under the exposure-targeted utility, denoted by [MATH] , an author’s utility is the number of impressions her document receives. Formally, |
Definition 1 (Exposure-targeted utility) The exposure-targeted utility of author [MATH] under a strategy profile [MATH] is given by |
[EQUATION] Note that [MATH] depends solely on the user mass of the topic she writes on and the probability of the mediator displaying her document. The other utility function is the action-targeted utility, denoted by [MATH] |
Definition 2 (Action-targeted utility) The action-targeted utility of author [MATH] under a strategy profile [MATH] is given by [EQUATION] |
Namely, an author’s utility is the user mass of her selected topic times the probability she is ranked first times the quality of her document. |
Overall, an authors game can be represented as a tuple [MATH] It is convenient to quantify the following; given a strategy profile [MATH] , let |
[MATH] denote the highest quality of a document on topic [MATH] , i.e., [EQUATION] Moreover, we denote by [MATH] the number of authors whose documents have the highest quality among those who write on topic [MATH] under [MATH] |
[EQUATION] Unless stated otherwise, we analyze games with a particular mediator, which is based on the PRP. Since we restrict the ranking list to include one rank only, the PRP coincides with ranking first the highest quality document on that topic. We denote by [MATH] the mediator that displays the document with the h... |
Definition 3 (The PRP Mediator) Given a quality matrix [MATH] , a topic [MATH] and a strategy profile [MATH] , the [MATH] ranks first the document of each author [MATH] with a probability of |
[EQUATION] 2.1 Further Game Theory Notation We now informally introduce some basic game theory concepts used throughout this paper. For an action profile [MATH] , we denote by [MATH] the action profile of all authors except author [MATH] A strategy [MATH] is called a better response of author [MATH] w.r.t. a strategy p... |
Given a strategy profile [MATH] , an improvement step is a profile [MATH] such that [MATH] is a better response of author [MATH] w.r.t. [MATH] An improvement path |
[MATH] is a sequence of improvement steps, where the improvements can be performed by different authors. Namely, in any improvement step along the improvement path exactly one author deviates from the strategy she selected in the previous step, but different authors can deviate in different steps. When the path [MATH] ... |
2.2 An Illustrative Example To further clarify our notation and setting, we provide the following example. Consider a game with [MATH] authors, [MATH] topics, a query distribution mass [MATH] such that [MATH] , a quality matrix |
[EQUATION] and [MATH] as the mediator. Given the utility function, the induced game can be viewed as a normal form bi-matrix game, as presented in Figure |
First, consider the exposure-targeted utility function. Consider the strategy profile [MATH] . Under this strategy profile the two authors write on topic 2, and their quality on that topic is the same, i.e., [MATH] ; thus, [MATH] and |
[EQUATION] Notice that author 2 can improve her utility by deviating to topic 1, i.e., to the strategy profile [MATH] . Indeed, this is an improvement step w.r.t. [MATH] . In this case, her utility is [MATH] . Clearly [MATH] is a PNE of this game. |
The action-targeted utility function induces a different bi-matrix game. The reader can verify that under this utility scheme, the unique PNE is [MATH] |
Better-Response Learning with the PRP Mediator In this section we show that under the PRP mediator, every better-response dynamics converges to a PNE, for both utility schemes. To make this claim more concrete, we use the following definition. |
Definition 4 We say that a mediator [MATH] is [MATH] -learnable if every game induced by [MATH] and the utility function [MATH] has the FIP property. |
Clearly, if any game that consists of [MATH] has the FIP property, then the authors can learn a PNE using any better-response dynamics. We use the above definition to crystallize our goals for this section: we wish to show that [MATH] is both [MATH] -learnable and [MATH] -learnable. Namely, in Subsection 3.1 we show th... |
Before we go on, we claim that the class of games induced by the PRP mediator does not have an exact potential. Proposition 1 The class of games induced by [MATH] and either one of [MATH] or [MATH] does not have an exact potential. |
Proof sketch of Proposition We show that the necessary condition for the existence of an exact potential ( ) does not hold for a general authors game with [MATH] authors. This result is obtained for both utility schemes. |
As mentioned in Section above, showing the convergence of any better-response dynamics in the lack of exact potential is challenging, and is nevertheless our goal for the rest of this section. In light of that, we shall introduce a further notation. |
Definition 5 Given a finite improvement path [MATH] , we define [EQUATION] i.e., [MATH] is the minimal number of authors writing documents with the highest quality on topic [MATH] |
Note that the minimum is taken over all steps in [MATH] 3.1 Exposure-Targeted Utility We now focus on games with [MATH] and [MATH] , namely the PRP mediator and the exposure-targeted utility function. We show that every improvement path is finite, suggesting that any better-response dynamics converges. The proof of thi... |
The following Proposition claims that in every improvement step, the improving author writes with a quality of at least the highest quality obtained in the preceding improvement step, on that particular topic. |
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