Locally Perturbed Random Walks with Unbounded Jumps

In \cite{SzT}, D. Sz\'asz and A. Telcs have shown that for the diffusively scaled, simple symmetric random walk, weak convergence to the Brownian motion holds even in the case of local impurities if $d \ge 2$. The extension of their result to finite range random walks is straightforward. Here, however, we are interested in the situation when the random walk has unbounded range. Concretely we generalize the statement of \cite{SzT} to unbounded random walks whose jump distribution belongs to the domain of attraction of the normal law. We do this first: for diffusively scaled random walks on $\mathbf Z^d$ $(d \ge 2)$ having finite variance; and second: for random walks with distribution belonging to the non-normal domain of attraction of the normal law. This result can be applied to random walks with tail behavior analogous to that of the infinite horizon Lorentz-process; these, in particular, have infinite variance, and convergence to Brownian motion holds with the superdiffusive $\sqrt{n \log n}$ scaling.Comment: 16 page

A mechanical model of normal and anomalous diffusion

The overdamped dynamics of a charged particle driven by an uniform electric field through a random sequence of scatterers in one dimension is investigated. Analytic expressions of the mean velocity and of the velocity power spectrum are presented. These show that above a threshold value of the field normal diffusion is superimposed to ballistic motion. The diffusion constant can be given explicitly. At the threshold field the transition between conduction and localization is accompanied by an anomalous diffusion. Our results exemplify that, even in the absence of time-dependent stochastic forces, a purely mechanical model equipped with a quenched disorder can exhibit normal as well as anomalous diffusion, the latter emerging as a critical property.Comment: 16 pages, no figure

What is relevant in hyperthermia treatment: heat, temperature, field or something else?

There are intensive discussions in the scientific communities on the quality-parameters of hyperthermia. Most of the parties are convinced that only the temperature decides about the optimal treatment, but strong doubts are also exists declaring the delivered heat (absorbed energy) or applied field (electro-magnetic influence) as primary effects. Strong points of the temperature-supporters are the results of all the investigations which are concentrating on the temperature-dependence and the effectivity of Arrhenius analysis showing a phase transition at about 42.5 C, (this is the basic of the step-down heating). On the other hand no doubts about the strong heat-dose dependence, which is most trivial by the treatment-time relevance in the clinical and laboratory results. The field effects in the hyperthermia does not investigated so widely, but trivial questions arise to choose the techniques, different applicators, frequencies and couplings. We had developed a set of the hyperthermia treatments applying various mixes of the heat and field effects. Four methods are applied successfully, where the heat- and field-effects are involved by different ratio for various malignant cases. Among the four methods there are a method with only electric field application (ECT, percutane, no heat), moderate heat with moderate field (PCT, cavitational), dominate local heat and electric field (EHY, loco-regional) and almost no field, only heat (WBH, whole-body application). The reached temperature in the tissue does not determined only by the absorbed heat, physically other effects have decisional role in the measured temperature. The applied power in the various methods is very different, ranging from a few watts to the few kW, while the electric field ranges from a few tens [mV/cell] to the few [nV/cell]. Due to the large and essential inhomogenities of the well-developed tumors only the average temperature could be defined in most of the cases. Same average temperature is reachable by different applied power, only depending on the power-delivery conditions. This observation strongly supports the importance of the heat (energy) absorption [W/kg] in the tumor instead of the only temperature conception. Results show the excellence of the optimizing the method to the actual case. With all the four treatment modalities we have surprisingly good results by choosing the most optimal for the given case and combining the applied methods with each other. Results by the different electro-hyperthermia applications will be shown in the presentation. The loco-regional applications are massively applied in deep-organ treatments (pancreas, liver, brain, etc.), while the cavitational is devoted for prostate, bladder and gynecological, etc. cases. The percutane local treatment is very effective for mammary-carcinomas, for head- and neck-cases, malignant melanomas, etc. The IR-A radiative whole-body hyperthermia is very useful in metastasizes, in systematic cases, etc. According to the intensive laboratory and clinical experiences and strong evidences we can conclude, that beside the temperature the heat-delivery as well as the field effects are primary important for the oncological hyperthermia