Scale Free Cluster Distributions from Conserving Merging-Fragmentation Processes

We propose a dynamical scheme for the combined processes of fragmentation and merging as a model system for cluster dynamics in nature and society displaying scale invariant properties. The clusters merge and fragment with rates proportional to their sizes, conserving the total mass. The total number of clusters grows continuously but the full time-dependent distribution can be rescaled over at least 15 decades onto a universal curve which we derive analytically. This curve includes a scale free solution with a scaling exponent of -3/2 for the cluster sizes.Comment: 4 pages, 3 figure

Diffusion, Fragmentation and Coagulation Processes: Analytical and Numerical Results

We formulate dynamical rate equations for physical processes driven by a combination of diffusive growth, size fragmentation and fragment coagulation. Initially, we consider processes where coagulation is absent. In this case we solve the rate equation exactly leading to size distributions of Bessel type which fall off as $\exp(-x^{3/2})$ for large $x$-values. Moreover, we provide explicit formulas for the expansion coefficients in terms of Airy functions. Introducing the coagulation term, the full non-linear model is mapped exactly onto a Riccati equation that enables us to derive various asymptotic solutions for the distribution function. In particular, we find a standard exponential decay, $\exp(-x)$, for large $x$, and observe a crossover from the Bessel function for intermediate values of $x$. These findings are checked by numerical simulations and we find perfect agreement between the theoretical predictions and numerical results.Comment: (28 pages, 6 figures, v2+v3 minor corrections

Aggregation with Multiple Conservation Laws

Aggregation processes with an arbitrary number of conserved quantities are investigated. On the mean-field level, an exact solution for the size distribution is obtained. The asymptotic form of this solution exhibits nontrivial ``double'' scaling. While processes with one conserved quantity are governed by a single scale, processes with multiple conservation laws exhibit an additional diffusion-like scale. The theory is applied to ballistic aggregation with mass and momentum conserving collisions and to diffusive aggregation with multiple species.Comment: 18 pages, te

Use of autologous bone graft in anterior cervical decompression: morbidity & quality of life analysis

Background: Autologous iliac crest graft has long been the gold standard graft material used in cervical fusion. However its harvest has significant associated morbidity, including protracted postoperative pain scores at the harvest site. Thus its continued practice warrants scrutiny, particularly now that alternatives are available. Our aims were to assess incidence and nature of complications associated with iliac crest harvest when performed in the setting of Anterior Cervical Decompression (ACD). Also, to perform a comparative analysis of patient satisfaction and quality of life scores after ACD surgeries, when performed with and without iliac graft harvest. Methods: All patients who underwent consecutive ACD procedures, with and without the use of autologous iliac crest graft, over a 48 month period were included (n = 53). Patients were assessed clinically at a minimum of 12 months postoperatively and administered 2 validated quality of life questionnaires: the SF-36 and Cervical Spine Outcomes Questionnaires (Response rate 96%). Primary composite endpoints included incidence of bone graft donor site morbidity, pain scores, operative duration, and quality of life scores. Results: Patients who underwent iliac graft harvest experienced significant peri-operative donor site specific morbidity, including a high incidence of pain at the iliac crest (90%), iliac wound infection (7%), a jejunal perforation, and longer operative duration (285 minutes vs. 238 minutes, p = 0.026). Longer term follow-up demonstrated protracted postoperative pain at the harvest site and significantly lower mental health scores on both quality of life instruments, for those patients who underwent autologous graft harvest Conclusion: ACD with iliac crest graft harvest is associated with significant iliac crest donor site morbidity and lower quality of life at greater than 12 months post operatively. This is now avoidable by using alternatives to autologous bone without compromising clinical or technical outcome