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Information generating, sharing and manipulating Source-Reservoir-Sink model of self-organizing dissipative structures

Information-theoretic description of the signal transmitter, the channel and receiver is extended to the network of self-organizing dissipative structures consisting of a source, a reservoir and a sink. The information generation by the source is subjected to controlled manipulation by the reservoir before being transmitted to the sink. The reservoir can have memory and variable capacity for information storage. The role of the reservoir in building the manipulative capacity for information storage and selective sharing is illustrated by the characteristic of asymmetric exchange between the reservoir and the sink. A box-model is used to develop the model to represent material, process and information sharing among the source, the reservoir and the sink. The model is applied to self-organizing carbon cages with the end-directed evolution of the buckyball.


I. INTRODUCTION
Emergence of self-organizing dissipative structures in a variety of energy consuming, entropy generating open systems has been described by a large number of eminent scholars, only a brief selection of the few references, relevant to the work described here, is given as refs. [1][2][3][4][5][6][7][8][9][10] and the references therein. This communication has emerged out of the investigations, experimental and theoretical, on various mechanisms of formation and fragmentation of carbon nanostructures. The mechanisms of formation of fullerenes in the soot generated by energetic heavy ion bombardments of graphite [11][12][13] and in the regenerative soot of hollow graphite discharges [14][15][16][17] have intrinsic similarities with the processes of fragmentation of the irradiated C60-fullerite [18] and carbon nanotubes [19]. With the help of the model developed here, we show that the energy dissipation and entropy generation during the formation and fragmentation stages can be described by the information-theoretic tools [20][21][22][23][24][25][26]. Two persistent themes have been noticed; (a) Maximum Entropy Production [27][28][29] may not be associated with the self-organizing, emergent structures, and (b) the inter-and intra-structural proesses play significant roles. After developing the general principles of the Source-Reservoir-Sink (SRS) model, we focus on the emergence of the most symmetric carbon cage-the Buckyball [30]. It emerges out of the selforganizing soot that contains carbon atoms, molecules, clusters and closed cages. The closed cages initially are formed in all shapes, sizes and isomeric densities. The cage transformations through fragmenting sequences and intra-cage bond reorientations [31] produce the icosahedral C60. It has been described as an 'order out of disorder' transition of self-organizing dissipative structures [32].
Kinetic and thermodynamic arguments to describe this process have had varying degrees of success [33][34][35][36][37]. In this communication, the far from equilibrium, irreversible dissipative structures are monitored through their information generation during the spatial and temporal stages, mostly within the Reservoir of the SRS model.
Shannon entropy is evaluated for each constituent of the irreversible dissipative structure that participate in the formative or fragmenting processes. Our model has the same basic design as that of Shannon's three-component system that is composed of signal Transmitter, the Channel and the signal Receiver. The Source symbolizes the Transmitter, Resevoir is equivalent to the Channel and Sink is the Receiver in our SRS model. Three significant modes of the model' application are (a) free-flow, (b) source-friendly and (c) the reservoir-specific modes are identified.
The source is the fountainhead where the process begins with the generation of the material and the information. The reservoir receives the material and information from the source at certain rates. Depending upon the design, the style, capacity and modes of communication with the sink, the material and information-sharing between the reservoir of information and the sink or the receiver occurs. The sink, in our model, is the sole recipient of the transfer of material, the associated processes and information from the reservoir. It is the end-directed evolution of the dissipative dynamical system of SRS. The transfer rate is determined by the mutual interconnectivity and the dynamical system-specific parameters. We will first present the general features of the SRS model and then in the second half of the paper, a specific application of the model will be discussed. The basic difference between our model and Shannon's description of a Source-Channel-Receiver is that our reservoir's ability of manipulation, enhancement or degradation of the information (or material) received from the source is a desireable feature unlike the noise generated by the Channel in Shannon model. Furthermore, the reservoir can have memory, fixed or variable. The reservoir may also bilaterally interact and exchange information (or material) with the sink. Our description of such a dissipative structure of source-reservoir-sink can display self-organisational character. The information-theoretic parameters are derived from Shannon entropy calculated for each constituent and the relative entropies between the constituents of the SRS model. We derive the entropic cost of the evolution of the emergence as a diagnostic tool. The information or fractal dimension of all SRS constituents categorizes the dissipative character of the end-directed evolution [38].

II. THE MODEL
The SRS model is an effort to start from the basics and build the foundation for understanding the self-organizational routes undertaken by certain dissipative structures. Our focus is the emergence of order out of disorder. The reverse process of the disorder generated in ordered structures like the single-walled carbon nanotubes has recently been discussed with similar information-theoretic arguments elsewhere [26]. After providing a general description of the SRS model, we will discuss the evolution of ensembles of carbon cages out of the disorder of hot carbon vapor. The model will be used to describe the emergence of the most symmetric cage-Buckyball from ensembles of fullerenes of all sorts of shapes, sizes and isomeric variations. In this case, the icosahedral C60 is the end-directed evolution of the self-organizing, dissipative dynamical system.
The first assumption of the model is that the flow or the exchange of material between the source reservoir and the sink can be discretized as pulses defined by the exchange rates. For example, in the case of two boxes connected by a controllable valve, the material can be transferred at a well-defined rate. If box#1 had 512 marbles or 512 liters of liquid, it can be exchanged with the box#0 at the rate of 1/2 by sharing 256, 128, 64, 32, 16, 8, 4, 2 and 1 in ten steps. At this stage, the time duration between pulses are not considered. The transferred quantity is the main parameter.
The probability paradigm is at the heart of the model. It defines the dissipative structure, the relative performance of its interactive constituents and displays the emergent characteristics of the modified or new structures. The normalized probability mass distributions for the source, reservoir and the sink are evaluated for different rates of flow and for various combinations thereof.
The probability for a set of distributive stages is referred to as ( ). The uncertainty is defined and measured as ln (1 ( )  ⁄  ). This function has special characteristics and has also been named 'surprise'. Hence the Kolmogrov measure of uncertainity [23] = ln (1 ( ) ⁄ ) (1).
The special characteristics associated with are related with the fact that at low ( ) its value increases. The sharply increasing rate of has the surprise [22]. We will illustrate this aspect in the section III.
Entropy can be evaluated from the product ( )ln (1 ( ) ⁄ ). The sum over all of this product is the well-known Shannon entropy or the information I It must be pointed out that and are dependent upon the measure and hence the flow rate.
Different configurations of the system (SRS) will yield varying and corresponding to the various rates of flow and the increasing or decreasing number of distributive stages .
Relative entropy is calculated for the dynamical systems to provide a measure of the Kuulback-Leibler [39] distance between two probability measures like ≡ ( ) and ≡ ( ) which are the probability distributions of the source and the sink, respectively. Relative probability is evaluated as The fourth diagnostic tool is the fractal dimension. The summation over every sequence of steps or stages, generates information ≡ ∑ ( )ln (1 ( ) ⁄ ), is the net information generated by the probability distribution ( ) ≡ ( ); where = , for respective probability distributions of dource, reservoir or the sink.
Fractal analysis based on I, for various flow rate configurations will be shown to be an indicator of the impact of the different phases or the modes of flow of the dissipative structures. Fractal or the information dimension is defined as [40][41][42] In equation (4), is the number of distributive stages or the measure required to obtain information.
The fractal dimension can be of the reservoir, the source or the sink; similar notation is used for all three components of SRS model.
An information-theoretic parameter is defined to provide an entropic cost measure of the emergence of the end-directed dissipative structure. It is the ratio of the total information generated by all the constituents ∑ to that generated by the emergent one as

III. FREE-FLOW MODE: SOURCE AND SINK ONLY
Let us consider a two-component model of a source and a sink, without the reservoit and hence no channel in Shannon's description of an information-network. In this, simplified Source and the Sink model, we ignore the material or the information lost in the transfer. The 2-Box version of the model assumes that the information generated by the Source or Box#1 is received at the pre-determined rate by the Box#0, without loss, manipulation of modification. We intend to develop the initial, essential information theoretic parameters that are cruicial for the understanding of the role of the reservoir in detail, that will be introduced in the next section. It will be shown that the Reservoir emerges as the manipulator of the original signal or the initial impetus given to the dynamical system.
Let us look at the probability distribution of the quantity of material shared between the source and the sink in Figure 1. With a fixed amount of material, the Source empties and the Sink being the sole recipient receives. The emptying of the Source and filling of the Sinks occurs at a pre-determined rate. The starting probabilities for the Source and the Sink are 1 and 0 at = 0.
The respective final probabilities for the two are ~0 and ~1 as → . (c) has the same two parameters for the Sink that is filling. The arrows show the directions of emptying and filling. (d) A random number generated between 0 and 1 is multiplied by 1/8 determines the flow from Source to the Sink. The emptying/filling range is extended to about twice as compared to that of figure 1. The randomness of the choice of the flow rate is reflected in the ln(1 /p( )) and entropy graphs in (e) and (f).

IV. THE SOURCE-RESERVOIR-SINK (SRS) MODEL
If a third box is added between the source and the sink, then a completely different flow dynamics emerges. Figure 2 shows the probabilities of the quantities of the material for the rate of flow @1/2 from the Source and @1/2 from the Reservoir to the Sink. These are the similar rates that were used in Figure 1 for the 2-Box model. through the Reservoir. In information theory, the Channel can be the source of noise, degradation and alteration of the original signal. In our model too, the Channel or the Resevoir plays the similar, but desireable, manipulable role. It is the repository of information, material or ideas. Some of its features may need to be controlled for desirable effects. We investigate these aspects where a dissipative dynamical system self-organizes and new structures emerge out of the interactions amongst the ingredients of the material received in the reservoir. An end-directed evolution of an emergent structure may occur that can be categorized using the diagnostic tools discussed in section I.   The relative entropies of two probability distributions that are connected by intermediary stage of a Reservoir, defined as the Kulbach-Leibler distance. In our case we denote these distributions as 2 ≡ and 0 ≡ . The relative entropy between the Source and Sink, from equation (3) ( 2 ∥ 0 ) = ∑ 2 ( ) ( 2 ( ) 0 ( ) ⁄ ) and the relative entropy of the distribution of the Sink with the Source is ( 0 ∥ 2 ) = ∑ 0 ( ) ( 0 ( ) 2 ( ) ⁄ ). These have been tabulated in Table I Table I. Only in the case of the 2-Box model ( 1 ∥ 0 ) < ( 0 ∥ 1 ); due to the absence of the reservoir.

V. THE CASE OF MULTIPLE SOURCES AND THE EXTENDED RESERVOIR
In the preceding section, the essential characteristics of an idealized SRS model based on 3 boxes, each representing one of the constituents were presented. In real physical situations, there can be more than one sources and reservoirs. Even the simplest environmental example of a lake feeding on waters from the surrounding hills and draining to a river, contains multiple sources of water. There can also be more than one lakes and rivers. One needs to deal with multiple Sources   Table   2 that this cost is higher for more effective, more manipulative and end-directed dissipative dynamical systems. For real physical systems, the entropic cost obtained by the technique described in the previous sections, is the minimum for the respective systems. Actual entropic cost is likely to be much higher due to multiple information exchanges between various constituents.
In situation where more than one outcomes are possible, then the cost may vary from system to sytem and the ambient conditions of multiple trasitions.

VII. THE CASE OF THE EMERGENCE OF THE BUCKYBALL-C 60
We apply the The driving forces of fragmentation are related to the inherent, structural instability due to the pentagon-related internal stresses and the degeneracy pressures of the Fermi gas of pi electrons [43]. Hot carbon vapor in an appropriately confined environment can generate fullerenes with relative densities that are determined by their respective heats of formation and isomeric abundance. Various mechanisms for the emergence of C60 out of the disorder of the hot C-vapor have been proposed ever since the discovery of Buckyball. Here we treat the grand canonical ensemble of the forming and fragmenting fullerenes as a dissipative dynamical system. By treating fullerenes as 3D rotors, a thermodynamic model was developed earlier with evaluation of thermal entropy of cage-to-cage transformations [37].   . Figure 12 is derived from the the information data for the fragmenting, constituent fullerenes of the ensembles and the emerging C60. The main objective of Figure 13 is to clarify some of the concepts that were introduced in this communication. It is also intended to suggest two basic points. (i) The first deals with the complex structures that emerge due to the self-organizing sub-structures or the components as the basic ingrdients. Self-organization requires energy input and appropriate environment. As we are dealing with carbon cages, therefore we assume this environment be the carbon vapor generated by laser ablation of graphite in vacuum. The constituents of this vapor will be the components and subsystems of Figure 13. This stage generates carbon cages of all shapes, sizes, isomeric variations and with pentagonal curvature-induced structural imperfections. We had reported earlier experimental results of such an evolution of carbon cages under high energy, heavy ionic irradiations of amorphous graphite where fullerenes ensembles consisted of the cages containing from ~30 to 90 carbon atoms [11][12][13]. These experiments were at room temperature where the sputtered atoms and molecules assembled into cages. These were not further transformed due to the lack of appropriate physical conditions and environment. It could be considered the 'stunted' growth stage of the pre-self-organizing ensembles of carbon cages. Another classic example is that of the pre-Buckyball discovery experiments of Rohlfing et al [44]. They had ablated graphite with laser and mass analyzed the emerging carbon 'clusters'. They did not recognize the importance of Cx>C30 as the closed cages-the fullerenes. Nor did they provide the environment of selforganization. Kroto and Smalley recognized the importance of cage-transformations as a result of the variations in the design of the extractor cone, pulses of He and other physical parameters [30].
In the context of our previous discussion, Rohlfing experiment falls in the category of the 'stunted growth' stage, like our experiments [11][12][13]. (ii) The second point deals with the next stage where these initially formed cages are the components and the sub-systems. The self-organization requires that the hot carbon atoms, molecules and cage-containing vapor continue the collision sequences of cage-cage, molecules-cage, molecules-molecules and of course, the collisions of all species with the cooling gas (He or Ar) and with the confining walls of the expansion chamber. It is here that the second stage of cage-transformations, discussed in detail earlier, begins. The cumulative information of the cage transformation processes provides clues to the evolution of complex structures. Self-organization is not a one-way process of evolution; involution also operates on the system such that the reverse sequences may also be operating to optimize the evolving structures like 2 + → +2 . The figure schematically describes these stages of involution and evolution.

VIII. CONCLUSIONS
An information-theoretic model has been presented that has its general features and as a special 5. We have shown that the dissipative dynamical system of fragmenting and transforming cages has an end-firected evolution towards C60. It acts as the sink whose basin is the entire ensemble.
6. Information is the crucial link between the phase space of a dissipative dynamical system and fractal dimensions of its constituents. It helps us to identify the processes of selforganization. . It describes graphically the generation of information, instantaneous and total by the ensemble of fragmenting fullerenes. The information is in the form of probabilities and entropies. A typical ensemble of 11 fullerenes from C60 to C80 represented by ∑ 80 60 and shown in the extreme left ellipse, is chosen as the example. The number densities of ensemble's fullerenes are proportional to their isomeric densities. Each, non-icosahedral fullerene, is subjected to the internal cage-fragmentation forces described in Figure 8. The ensemble is subjected to the hot sooty environment of the condensing carbon vapor that results in the aftermath of laser ablative pulses. The model assumes fragmentation of half of all fullerenes at each fragmentation stage . The larger fragmenting cages populate the smaller cage densities as described in section V and displayed graphically in Figure 7.